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

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> ;<br/>
<a NAME="E1:128_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K14" target="_self">*'</a> <font color="Maroon">J</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E3:128_1_1"/>
for <font color="Olive">y</font>, <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:128_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then 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">R</font><b> such that </b><br/><a NAME="E3:128_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) ) )
 ;<br/>
<b>set </b><font color="Maroon">q</font> = <font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font>;<br/>
<a NAME="E4:128_1_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binom.html#T4">BINOM:4</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <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"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</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="relset_1.html#K4">dom</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
 <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 NAME="E6:128_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1_2"><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="E1:128_1_1_1_2"/><i><font color="Green">E72</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">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span> )
 ;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">r'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:128_1_1_1_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">r'</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">r'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:128_1_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <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> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</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"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ideal_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:128_1_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</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 NAME="E6:128_1_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">r'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green">E79</font></i>: <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">r'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ideal_1.html#T13" target="_self">Th13</a>, <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><b>thus </b><a NAME="E8:128_1_1_1_2"/>
 ex <font color="Olive">b</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">r</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font>
;<br/>

<b>take </b>
<font color="Maroon">r'</font>
;<br/>

<b>thus </b><a NAME="E1:128_1_1_1_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">r'</font> &amp; <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">r'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 <b>by </b><i><a class="ref" href="ideal_1.html#D5" target="_self">Def5</a>, <a class="ref" href="ideal_1.html#T14" target="_self">Th14</a>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:128_1_1_1"/>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:128_1_1"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K14" target="_self">*'</a> <font color="Maroon">J</font> is <a href="ideal_1.html#V2" target="_self">left-ideal</a>
 <b>by </b><i/>;<br/>


</div>
