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

<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="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1">add-cancelable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> ;<br/>
<a NAME="E1:90"/><i><font color="Green">E41</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> ;<br/>
<a NAME="E1:90_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">lc</font> be    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font>;<br/>

<b>assume </b><a NAME="E1:90_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:90_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:90_1_1"/><b>then </b>
<font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E4:90_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</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="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font>;<br/>
<a NAME="E6:90_1_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:90_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:90_1_1"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</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="binom.html#T1">BINOM:1</a></i>;<br/>
<b>hence </b><a NAME="E9:90_1_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:90_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:90_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>thus </b><a NAME="E2:90_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">lc</font> be    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font>;<br/>

<b>assume </b><a NAME="E1:90_1_2_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_1_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="ideal_1.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:90_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_2_1_1_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:90_1_2_1_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:90_1_2_1_1_2"/><b>then </b>
not <font color="Maroon">lc</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font>, <font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:90_1_2_1_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> is    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="ideal_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E6:90_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E7:90_1_2_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:90_1_2_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E9:90_1_2_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E11:90_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:90_1_2_1_1_2"/><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green">E121</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> = 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="binom.html#T3">BINOM:3</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="ideal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E14:90_1_2_1_1_2"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i/>;<br/><a NAME="E15:90_1_2_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E16:90_1_2_1_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E17:90_1_2_1_1_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>;<br/><b>hence </b><a NAME="E18:90_1_2_1_1_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T6" target="_self">Th6</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<a NAME="E3:90_1"/><i><font color="Green">E123</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="ideal_1.html#T2" target="_self">Th2</a>, );<br/></i>
<b>assume </b><a NAME="E4:90_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:90_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 ;<br/>
<b>consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E8:90_1"/><i><font color="Green">E125</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E9:90_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font>]
 
<b>by </b><i><a class="ref" href="ideal_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E10:90_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T4" target="_self">Th4</a>, <a class="ref" href="ideal_1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:90"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:90_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">lc</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:90_2_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:90_2_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</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/><a NAME="E3:90_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E4:90_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="ideal_1.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/><b>hence </b><a NAME="E6:90_2_1"/>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> st <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">lc</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> <b>by </b><i/>;<br/>
<a NAME="E5:90_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="binom.html#T3">BINOM:3</a></i>;<br/>
<b>hence </b><a NAME="E6:90_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:90"/><i><font color="Green">E131</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:90_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:90_3"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:90_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:90"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</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">L</font> <b>by </b><i><a class="ref" href="ideal_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="ideal_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I'</font> = <font color="Maroon">I</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E8:90"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">I'</font> is <a href="ideal_1.html#V1" target="_self">add-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D1" target="_self">IDEAL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:90_4"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I'</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:90_4"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 ;<br/>
<b>consider </b><font color="Maroon">y'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:90_4"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> )
 <b>by </b><i><a class="ref" href="ideal_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>consider </b><font color="Maroon">lcx</font> being    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E7:90_4"/><i><font color="Green">E139</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lcx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">lcy</font> being    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E9:90_4"/><i><font color="Green">E140</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lcy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:90_4"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">lcx</font> <a href="ideal_1.html#K5" target="_self">^</a> <font color="Maroon">lcy</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>;<br/>
<b>hence </b><a NAME="E11:90_4"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I'</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:90"/>
<font color="Maroon">I'</font> is <a href="ideal_1.html#V2" target="_self">left-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D2" target="_self">IDEAL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:90_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:90_5"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 ;<br/>
<b>consider </b><font color="Maroon">lcx</font> being    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E5:90_5"/><i><font color="Green">E142</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lcx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">plcx</font> = <font color="Maroon">p</font> <a href="polynom1.html#K6">*</a> <font color="Maroon">lcx</font> as    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> <b>by </b><i/>;<br/>
<a NAME="E7:90_5"/>
<font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">plcx</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="binom.html#T4">BINOM:4</a></i>;<br/>
<b>hence </b><a NAME="E8:90_5"/>
<font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:90"/><b>then </b>
<font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T2" target="_self">Th2</a>, <a class="ref" href="ideal_1.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E11:90"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:90_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:90_6"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">lc</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="ideal_1.html#T5" target="_self">Th5</a></i>;<br/><b>thus </b><a NAME="E4:90_6"/>
 ex <font color="Olive">f</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">f</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E13:90"/>
 ex <font color="Olive">f</font> being   <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">f</font>
 ;<br/>

<b>hence </b><a NAME="E14:90"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T5" target="_self">Th5</a></i>;<br/>


</div>
