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

<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">r</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">cL</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<b>reconsider </b><font color="Maroon">pqra</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font>, <font color="Maroon">pqrb</font> = <font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">pq</font> = <font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">qr</font> = <font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">db</font> =  <a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font>;<br/><b>consider </b><font color="Maroon">ls</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:174_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">pqra</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">ls</font>
 <b>and </b><br/><a NAME="E3:174_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">ls</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E4:174_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ls</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</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">ls</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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/><b>consider </b><font color="Maroon">rs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:174_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">pqrb</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">rs</font>
 <b>and </b><br/><a NAME="E7:174_1"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">rs</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E8:174_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rs</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</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">rs</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" target="_self">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b> <a href="matrlin.html#V1">FinSequence-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> = <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>;<br/><b>consider </b><font color="Maroon">dbl</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E10:174_1"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">dbl</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E11:174_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font> holds <br/><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_5.html#S1">FINSEQ_5:sch 1</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b> <a href="matrlin.html#V1">FinSequence-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">dbr</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E13:174_1"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">dbr</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E14:174_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font> holds <br/><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_5.html#S1">FINSEQ_5:sch 1</a>();<br/></i><a NAME="E15:174_1"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbl</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:174_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbl</font>
 ;<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="E3:174_1_1"/><i><font color="Green">E108</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">dbl</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dbl</font> <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/>
<b>set </b><font color="Maroon">ddbk1</font> =  <a href="polynom1.html#K21" target="_self">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">dbk2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2;<br/>
<b>set </b><font color="Maroon">dblk</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>;<br/>
<a NAME="E4:174_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">dbl</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="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:174_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1"><i><font color="Green">E112</font></i></a></i>;<br/>
<i><font color="Green">E110</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>

;<br/>
<a NAME="E6:174_1_1"/><i><font color="Green">E111</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:174_1_1"/><i><font color="Green">E112</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:174_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:174_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
 ;<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="E3:174_1_1_2"/><i><font color="Green">E113</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"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:174_1_1_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E7:174_1_1_2"><i><font color="Green">E114</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">Fi</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><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="E7:174_1_1_2"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E7:174_1_1_2"><i><font color="Green">E114</font></i></a>, , , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Gi</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<a NAME="E9:174_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fi</font> <a href="binarith.html#K10">^</a> <font color="Maroon">Gi</font>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, <a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:174_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:174_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> is    <a href="finseq_1.html#M2">FinSequence</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:174_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="txt" href="polynom1.html#E7:174_1_1"><i><font color="Green">E112</font></i></a>, , <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:174_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbr</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:174_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbr</font>
 ;<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="E3:174_1_2"/><i><font color="Green">E115</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">dbr</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dbr</font> <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/>
<b>set </b><font color="Maroon">ddbk1</font> =  <a href="polynom1.html#K21" target="_self">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">dbk2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1;<br/>
<b>set </b><font color="Maroon">dbrk</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E4:174_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">dbr</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E117</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>

;<br/>
<a NAME="E6:174_1_2"/><i><font color="Green">E194</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:174_1_2"/><i><font color="Green">E195</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:174_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:174_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span>
 ;<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="E3:174_1_2_2"/><i><font color="Green">E196</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:174_1_2_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="polynom1.html#E5:174_1_2"><i><font color="Green">E117</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">Fi</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="txt" href="polynom1.html#E3:174_1_2"><i><font color="Green">E115</font></i></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="polynom1.html#E5:174_1_2"><i><font color="Green">E117</font></i></a></i>;<br/>
<a NAME="E7:174_1_2_2"/><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="txt" href="polynom1.html#E4:174_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="polynom1.html#E5:174_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Gi</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<a NAME="E9:174_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gi</font> <a href="binarith.html#K10">^</a> <font color="Maroon">Fi</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E5:174_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, <a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:174_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:174_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> is    <a href="finseq_1.html#M2">FinSequence</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:174_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <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/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">dbl</font> = <font color="Maroon">dbl</font>, <font color="Maroon">dbr</font> = <font color="Maroon">dbr</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <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/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="finseq_2.html#M2">Element</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="funct_2.html#NM1">Function</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E19:174_1"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">b</font> being   <a href="finseq_2.html#M2">Element</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E20:174_1"/><i><font color="Green">E202</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>set </b><font color="Maroon">fdbl</font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbl</font>;<br/><b>set </b><font color="Maroon">fdbr</font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbr</font>;<br/><b>reconsider </b><font color="Maroon">vfdbl</font> = <font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbl</font></span>)</span>, <font color="Maroon">vfdbr</font> = <font color="Maroon">v</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbr</font></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">L</font> <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/><b>consider </b><font color="Maroon">vdbl</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E23:174_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">vdbl</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">v</font></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">dbl</font>
 <b>and </b><br/><a NAME="E24:174_1"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">vfdbl</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">vdbl</font>
 <b>by </b><i/>;<br/><a NAME="E25:174_1"/><i><font color="Green">E206</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">vfdbl</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</font></span>)</span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_3"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">f</font> =  <a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</font>;<br/><a NAME="E1:174_1_3"/><i><font color="Green">E207</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">vdbl</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vdbl</font>
 <b>by </b><i/>;<br/><i><font color="Green">E208</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vdbl</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font>

;<br/><b>hence </b><a NAME="E3:174_1_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ls</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:174_1_3"/><i><font color="Green">E211</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ls</font> )
 ;<br/><a NAME="E5:174_1_3"/><i><font color="Green">E212</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ls</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, , , <a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E6:174_1_3"/><i><font color="Green">E213</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="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, , , <a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, <a class="ref" href="polynom1.html#T34" target="_self">Th34</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then 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="E8:174_1_3"/><i><font color="Green">E214</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</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>
 <b>and </b><br/><a NAME="E9:174_1_3"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">ls</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T30" target="_self">Th30</a>, <a class="ref" href="polynom1.html#T35" target="_self">Th35</a></i>;<br/><b>reconsider </b><font color="Maroon">b2'</font> = <font color="Maroon">b2</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">pqs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:174_1_3"/><i><font color="Green">E216</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">pqs</font>
 <b>and </b><br/><a NAME="E13:174_1_3"/><i><font color="Green">E217</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pqs</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" target="_self">decomp</a> <font color="Maroon">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E14:174_1_3"/><i><font color="Green">E219</font></i>: 
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#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pqs</font> holds <br/> ex <font color="Olive">b11</font>, <font color="Olive">b12</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" target="_self">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b11</font>,<font color="Olive">b12</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">pqs</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b11</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b12</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E15:174_1_3"/><i><font color="Green">E220</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">pqs</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E16:174_1_3"/><i><font color="Green">E222</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">vdbl</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, , <a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, <a class="ref" href="polynom1.html#T34" target="_self">Th34</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">vdblk</font> = <font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>;<br/><b>set </b><font color="Maroon">ddbk1</font> =  <a href="polynom1.html#K21" target="_self">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">dbk2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2;<br/><a NAME="E17:174_1_3"/>
 <a href="finseq_1.html#K3">len</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> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E18:174_1_3"/><b>then </b><i><font color="Green">E223</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E224</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
<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="funct_1.html#K1">.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><i><font color="Green">E225</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<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="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><a NAME="E21:174_1_3"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">dbl</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="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, </i>;<br/><b>set </b><font color="Maroon">dblk</font> = <font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/><i><font color="Green">E227</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D6" target="_self">Def6</a>, <a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>

;<br/><a NAME="E23:174_1_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a>, , <a class="ref" href="polynom1.html#T34" target="_self">Th34</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:174_1_3"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbl</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E25:174_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbl</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <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="E26:174_1_3"/><b>then </b>
<font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#T39" target="_self">Th39</a></i>;<br/><b>then reconsider </b><font color="Maroon">dblk</font> = <font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="finseq_1.html#M2">FinSequence</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E28:174_1_3"/><i><font color="Green">E258</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E29:174_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">dblk</font> <a href="tarski.html#R1">c=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E30:174_1_3"/><b>then </b><i><font color="Green">E259</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dblk</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E31:174_1_3"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">vdblk</font> = <font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E33:174_1_3"/><i><font color="Green">E260</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pqs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_3_5"><b>now </b></a><div class="add"><b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">vdblk</font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pqs</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D5" target="_self">Def5</a>, <a class="ref" href="polynom1.html#T38" target="_self">Th38</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
;<br/><a NAME="E2:174_1_3_5"/><b>then </b><i><font color="Green">E261</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vdblk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:174_1_3_5"/><i><font color="Green">E262</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>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E4:174_1_3_5"/><b>then </b><i><font color="Green">E263</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"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">b11</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b12</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:174_1_3_5"/><i><font color="Green">E265</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</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">b11</font>,<font color="Maroon">b12</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E7:174_1_3_5"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">pqs</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">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b11</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b12</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7" target="_self">Def7</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>  ;<br/><a NAME="E9:174_1_3_5"/><i><font color="Green">E267</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T38" target="_self">Th38</a>, , , <a class="ref" href="polynom1.html#T41" target="_self">Th41</a>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E10:174_1_3_5"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D5" target="_self">Def5</a>, <a class="ref" href="polynom1.html#T38" target="_self">Th38</a>, , <a class="ref" href="polynom1.html#T41" target="_self">Th41</a>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then </b><i><font color="Green">E268</font></i>: <font color="Maroon">dblk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b11</font>,<font color="Maroon">b12</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">b2</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D5" target="_self">Def5</a>, <a class="ref" href="polynom1.html#D6" target="_self">Def6</a>, , <a class="ref" href="polynom1.html#T41" target="_self">Th41</a>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="polynom1.html#T43" target="_self">Th43</a>, <a class="ref" href="polynom1.html#D9" target="_self">Def9</a>, <a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">b11</font>,<font color="Maroon">b12</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60">FINSEQ_1:60</a></i>
;<br/><b>reconsider </b><font color="Maroon">b11'</font> = <font color="Maroon">b11</font>, <font color="Maroon">b12'</font> = <font color="Maroon">b12</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">b11'</font>,<font color="Maroon">b12'</font>,<font color="Maroon">b2'</font></span><a href="finseq_4.html#K3">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T124">FINSEQ_2:124</a></i>;<br/><a NAME="E14:174_1_3_5"/><i><font color="Green">E269</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">pqs</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, <a class="ref" href="polynom1.html#D8" target="_self">Def8</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:174_1_3_5"/><i><font color="Green">E270</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pqs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">dblk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T41" target="_self">Th41</a>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b11'</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b11</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b12</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#T44" target="_self">Th44</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></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#T44" target="_self">Th44</a>, <a class="ref" href="polynom1.html#D11" target="_self">Def11</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div><a NAME="E35:174_1_3"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">vdblk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><i><font color="Green">E272</font></i>: <font color="Maroon">vdbl</font> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">vdbl</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">dbl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<font color="Maroon">pqs</font> <a href="polynom1.html#K7" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, , <a class="ref" href="polynom1.html#T41" target="_self">Th41</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E37:174_1_3"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">ls</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T35" target="_self">Th35</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E38:174_1_3"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</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">vdbl</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/><b>hence </b><a NAME="E39:174_1_3"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T36" target="_self">Th36</a>, , , <a class="ref" href="polynom1.html#D8" target="_self">Def8</a>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="matrlin.html#D8">MATRLIN:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E27:174_1"/><b>then </b><i><font color="Green">E274</font></i>: 
 <a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbl</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ls</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><b>consider </b><font color="Maroon">vdbr</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E29:174_1"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">vdbr</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">v</font></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">dbr</font>
 <b>and </b><br/><a NAME="E30:174_1"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">vfdbr</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">vdbr</font>
 <b>by </b><i/>;<br/><a NAME="E31:174_1"/><i><font color="Green">E278</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">vfdbr</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T34" target="_self">Th34</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_4"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">f</font> =  <a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</font>;<br/><a NAME="E1:174_1_4"/><i><font color="Green">E281</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">vdbr</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vdbr</font>
 <b>by </b><i/>;<br/><i><font color="Green">E282</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vdbr</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font>

;<br/><b>hence </b><a NAME="E3:174_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rs</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:174_1_4"/><i><font color="Green">E283</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rs</font> )
 ;<br/><a NAME="E5:174_1_4"/><i><font color="Green">E284</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E6:174_1_4"/><i><font color="Green">E287</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="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="polynom1.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then 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="E8:174_1_4"/><i><font color="Green">E288</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</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>
 <b>and </b><br/><a NAME="E9:174_1_4"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">rs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T32" target="_self">Th32</a>, </i>;<br/><b>reconsider </b><font color="Maroon">b1'</font> = <font color="Maroon">b1</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">qrs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:174_1_4"/><i><font color="Green">E290</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">qrs</font>
 <b>and </b><br/><a NAME="E13:174_1_4"/><i><font color="Green">E292</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">qrs</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" target="_self">decomp</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E14:174_1_4"/><i><font color="Green">E293</font></i>: 
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#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qrs</font> holds <br/> ex <font color="Olive">b11</font>, <font color="Olive">b12</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" target="_self">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b11</font>,<font color="Olive">b12</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">qrs</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b11</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b12</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E15:174_1_4"/><i><font color="Green">E294</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">qrs</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E16:174_1_4"/><i><font color="Green">E295</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">vdbr</font>
 <b>by </b><i>, , , <a class="ref" href="polynom1.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">vdbrk</font> = <font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>;<br/><b>set </b><font color="Maroon">ddbk1</font> =  <a href="polynom1.html#K21" target="_self">decomp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>;<br/><b>set </b><font color="Maroon">dbk2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1;<br/><a NAME="E17:174_1_4"/>
 <a href="finseq_1.html#K3">len</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> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E18:174_1_4"/><b>then </b><i><font color="Green">E296</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E297</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<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="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><i><font color="Green">E298</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
<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="funct_1.html#K1">.</a> 2
<b>by </b><i>, <a class="ref" href="polynom1.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><a NAME="E21:174_1_4"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">dbr</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="polynom1.html#K4" target="_self">^^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D6" target="_self">Def6</a></i>;<br/><b>set </b><font color="Maroon">dbrk</font> = <font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/><i><font color="Green">E302</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>

;<br/><a NAME="E23:174_1_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:174_1_4"/><b>then </b><i><font color="Green">E303</font></i>: 
<font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbr</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E25:174_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbr</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <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="E26:174_1_4"/><b>then </b>
<font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#T41" target="_self">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon">dbrk</font> = <font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="finseq_1.html#M2">FinSequence</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E28:174_1_4"/><i><font color="Green">E304</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E29:174_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">dbrk</font> <a href="tarski.html#R1">c=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E30:174_1_4"/><b>then </b><i><font color="Green">E306</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbrk</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E31:174_1_4"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">vdbrk</font> = <font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E33:174_1_4"/><i><font color="Green">E307</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qrs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E34:174_1_4"/><b>then </b><i><font color="Green">E310</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">qrs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E35:174_1_4"/><b>then </b><i><font color="Green">E313</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vdbrk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E36:174_1_4"/><i><font color="Green">E314</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vdbrk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="polynom1.html#T43" target="_self">Th43</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_4_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:174_1_4_5"/><i><font color="Green">E315</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>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span> )
 ;<br/><a NAME="E2:174_1_4_5"/><b>then </b><i><font color="Green">E316</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">dbrk</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">b11</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b12</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:174_1_4_5"/><i><font color="Green">E319</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</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">b11</font>,<font color="Maroon">b12</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:174_1_4_5"/><i><font color="Green">E320</font></i>: 
<font color="Maroon">qrs</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">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b11</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b12</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="polynom1.html#T43" target="_self">Th43</a>, <a class="ref" href="polynom1.html#D10" target="_self">Def10</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>  ;<br/><a NAME="E7:174_1_4_5"/><i><font color="Green">E321</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, <a class="ref" href="polynom1.html#D8" target="_self">Def8</a>, <a class="ref" href="polynom1.html#D11" target="_self">Def11</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E8:174_1_4_5"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7" target="_self">Def7</a>, <a class="ref" href="polynom1.html#D11" target="_self">Def11</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then </b><i><font color="Green">E322</font></i>: <font color="Maroon">dbrk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">b1</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b11</font>,<font color="Maroon">b12</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom1.html#T39" target="_self">Th39</a>, , , <a class="ref" href="polynom1.html#D11" target="_self">Def11</a>, , <a class="ref" href="polynom1.html#T45" target="_self">Th45</a>, <a class="ref" href="matrlin.html#D2">MATRLIN:def 2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b11</font>,<font color="Maroon">b12</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T60">FINSEQ_1:60</a></i>
;<br/><b>reconsider </b><font color="Maroon">b11'</font> = <font color="Maroon">b11</font>, <font color="Maroon">b12'</font> = <font color="Maroon">b12</font> as    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<font color="Maroon">b11'</font>,<font color="Maroon">b12'</font></span><a href="finseq_4.html#K3">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T124">FINSEQ_2:124</a></i>;<br/><a NAME="E12:174_1_4_5"/><i><font color="Green">E323</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">qrs</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T43" target="_self">Th43</a>, <a class="ref" href="polynom1.html#T44" target="_self">Th44</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">dbrk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T42" target="_self">Th42</a>, <a class="ref" href="polynom1.html#D11" target="_self">Def11</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D12" target="_self">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b11</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b11</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b12</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qrs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#T46" target="_self">Th46</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#T43" target="_self">Th43</a>, <a class="ref" href="polynom1.html#T46" target="_self">Th46</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div><a NAME="E38:174_1_4"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">vdbrk</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" target="_self">Def9</a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><i><font color="Green">E325</font></i>: <font color="Maroon">vdbr</font> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">vdbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="polynom1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</font></span>)</span> <a href="rfunct_3.html#K5">--&gt;</a> <font color="Maroon">v</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">dbr</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T33" target="_self">Th33</a>, <a class="ref" href="polynom1.html#D6" target="_self">Def6</a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K6" target="_self">*</a> <font color="Maroon">qrs</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D6" target="_self">Def6</a>, <a class="ref" href="polynom1.html#T44" target="_self">Th44</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E40:174_1_4"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">rs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs</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/><a NAME="E41:174_1_4"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</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">vdbr</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/><b>hence </b><a NAME="E42:174_1_4"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="polynom1.html#D5" target="_self">Def5</a>, <a class="ref" href="polynom1.html#D11" target="_self">Def11</a>, , <a class="ref" href="matrlin.html#D8">MATRLIN:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E33:174_1"/><b>then </b><i><font color="Green">E327</font></i>: 
 <a href="matrlin.html#K8">Sum</a> <font color="Maroon">vdbr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rs</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><a NAME="E34:174_1"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbl</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">dbr</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" target="_self">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/><b>then consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbl</font></span>)</span><b> such that </b><br/><a NAME="E36:174_1"/><i><font color="Green">E328</font></i>: 
 <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbr</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbl</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E37:174_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">dbl</font></span>)</span> <a href="tarski.html#R1">c=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E38:174_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vfdbl</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">dbl</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vfdbl</font> ;<br/><a NAME="E40:174_1"/>
<font color="Maroon">vfdbr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vfdbl</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/><b>hence </b><a NAME="E41:174_1"/>
<font color="Maroon">pqra</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pqrb</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T31" target="_self">Th31</a>, , , <a class="ref" href="polynom1.html#T35" target="_self">Th35</a>, , <a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:174"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
