<?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#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</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#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>









<b>set </b><font color="Maroon">q</font> = <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font>;<br/>
<b>consider </b><font color="Maroon">s</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="E2:18"/><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">s</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</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">m</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/>
<i><font color="Green">E24</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></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">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:18"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T73">POLYNOM1:73</a></i>;<br/>
<b>consider </b><font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:18"/><i><font color="Green">E27</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> &amp; <font color="Maroon">s</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">m</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/>
<i><font color="Green">E38</font></i>:  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font> = 
<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_1.html#T61">FINSEQ_1:61</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">E49</font></i>: <font color="Maroon">b2</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span>,<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_1.html#T61">FINSEQ_1:61</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="E10:18"/>
for <font color="Olive">k'</font> being   <a href="subset_1.html#M1">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">s</font> &amp; <font color="Olive">k'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_4"/><i><font color="Green">E50</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">s</font> &amp; <font color="Maroon">k'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <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="E3:18_4"/><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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> &amp; <font color="Maroon">s</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">m</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:18_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font>
 
<b>by </b><i>, , <a class="txt" href="polyred.html#E2:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>
<i><font color="Green">E53</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1'</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1'</font></span>)</span></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_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<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="txt" href="polyred.html#E2:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polyred.html#E3:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_4_2"/>
( ( <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font> &amp; <font color="Maroon">b2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> ) or <font color="Maroon">b1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font> or <font color="Maroon">b2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_4_2_1"><b>suppose </b></a><a NAME="E1:18_4_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font> &amp; <font color="Maroon">b2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</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"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="polyred.html#E2:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polyred.html#E5:18"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:18_4_2_1"/>
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_4_2_2"><b>suppose </b></a><a NAME="E1:18_4_2_2"/>
<font color="Maroon">b1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:18_4_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_4_2_2"/>
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E2:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_4_2_3"><b>suppose </b></a><a NAME="E1:18_4_2_3"/>
<font color="Maroon">b2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:18_4_2_3"/><b>then </b>
<font color="Maroon">b1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/><a NAME="E3:18_4_2_3"/><b>then </b>
<font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:18_4_2_3"/>
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="polyred.html#E2:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E11:18"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>;<br/>


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