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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V7" title="ALGSTR_0:attr.7">add-cancelable</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>set </b><font color="Maroon" title="c3">I</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span> ;<br/>
<a NAME="E1:89"/><i><font color="Green" title="E43">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> ;<br/>
<a NAME="E1:89_1"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">lc</font> be    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font>;<br/>

<b>assume </b><a NAME="E1:89_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:89_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E3:89_1_1"/><b>then </b>
<font color="Maroon" title="c5">lc</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E4:89_1_1"/><b>then </b><i><font color="Green" title="E45">A3</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c5">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font>;<br/>
<a NAME="E6:89_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def15</a></i>;<br/>
<a NAME="E7:89_1_1"/><b>then </b><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:89_1_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="binom.html#T1" title="BINOM:th.1">BINOM:1</a></i>;<br/>
<b>hence </b><a NAME="E9:89_1_1"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c5">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:89_1_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="ideal_1.html#E7:89_1_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:89_1"/><i><font color="Green" title="E45">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:89_1_2"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/>

<b>thus </b><a NAME="E2:89_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">lc</font> be    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font>;<br/>

<b>assume </b><a NAME="E1:89_1_2_1"/><i><font color="Green" title="E47">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_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="E1:89_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="ideal_1.html#E1:89_1_2_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:89_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c6">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:89_1_2"><i><font color="Green" title="E46">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:89_1_2_1_1_2"/><i><font color="Green" title="E48">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c6">lc</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">q</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c8">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:89_1_2_1_1_2"/><i><font color="Green" title="E49">A9</font></i>: 
( <font color="Maroon" title="c6">lc</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> )
 <b>by </b><i><a class="ref" href="ideal_1.html#T32" target="_self" title="IDEAL_1:th.32">Th32</a></i>;<br/><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ideal_1.html#E1:89_1_2_1_1_2"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="ideal_1.html#E4:89_1_2_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/><a NAME="E6:89_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">A10</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:89_1_2"><i><font color="Green" title="E46">A6</font></i></a></i>;<br/><a NAME="E7:89_1_2_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E8:89_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c11">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E10:89_1_2_1_1_2"/><i><font color="Green" title="E52">A12</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:89_1_2_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def9</a></i>;<br/><a NAME="E11:89_1_2_1_1_2"/><i><font color="Green" title="E53">A13</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="ideal_1.html#E8:89_1_2_1_1_2"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><i><font color="Green" title="E54">A14</font></i>:  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> = 
<font color="Maroon" title="c8">r</font>
<b>by </b><i><a class="ref" href="binom.html#T3" title="BINOM:th.3">BINOM:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">v</font>
<b>by </b><i><a class="txt" href="ideal_1.html#E10:89_1_2_1_1_2"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="ideal_1.html#E11:89_1_2_1_1_2"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><a NAME="E13:89_1_2_1_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def15</a></i>;<br/><a NAME="E14:89_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E15:89_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</a></i>;<br/><a NAME="E16:89_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E55">A15</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E12:89_1_2_1_1_2"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="ideal_1.html#D3" target="_self" title="IDEAL_1:def.3">Def3</a></i>;<br/><a NAME="E17:89_1_2_1_1_2"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c6">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:89_1_2_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>;<br/><b>hence </b><a NAME="E18:89_1_2_1_1_2"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c6">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:89_1_2_1_1_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="ideal_1.html#E16:89_1_2_1_1_2"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</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="E3:89_1"/><i><font color="Green" title="E46">A16</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="ideal_1.html#E1:89_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="ideal_1.html#E2:89_1"><i><font color="Green" title="E45">A5</font></i></a>);<br/></i>
<b>assume </b><a NAME="E4:89_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:89_1"/><i><font color="Green" title="E47">A17</font></i>: 
( <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  ex <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x'</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">lc</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E8:89_1"/><i><font color="Green" title="E48">A18</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c6">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:89_1"><i><font color="Green" title="E47">A17</font></i></a></i>;<br/>
<a NAME="E9:89_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">lc</font>]
 
<b>by </b><i><a class="txt" href="ideal_1.html#E3:89_1"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:89_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:89_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="ideal_1.html#E8:89_1"><i><font color="Green" title="E48">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:89"/><i><font color="Green" title="E44">A19</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_2"/><i><font color="Green" title="E45">A20</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>set </b><font color="Maroon" title="c6">lc</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:89_2_1"/><i><font color="Green" title="E46">A21</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><a NAME="E2:89_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:89_2_1"/><b>then </b>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:89_2_1"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
;<br/><b>hence </b><a NAME="E5:89_2_1"/>
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ex <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> st <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:89_2"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="ideal_1.html#E1:89_2_1"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">lc</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def9</a></i>;<br/>
<a NAME="E5:89_2"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c7">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="ref" href="binom.html#T3" title="BINOM:th.3">BINOM:3</a></i>;<br/>
<b>hence </b><a NAME="E6:89_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:89"/><i><font color="Green" title="E45">A22</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:89_3"/><i><font color="Green" title="E46">A23</font></i>: 
( <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  ex <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:89_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:89_3"><i><font color="Green" title="E46">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:89"/><i><font color="Green" title="E46">A24</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">I</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="ideal_1.html#E2:89"><i><font color="Green" title="E44">A19</font></i></a>, <a class="txt" href="ideal_1.html#E3:89"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="ideal_1.html#E5:89"><i><font color="Green" title="E46">A24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">I'</font> = <font color="Maroon" title="c5">I</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E8:89"/><i><font color="Green" title="E47">A25</font></i>: 
<font color="Maroon" title="c6">I'</font> is <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">IDEAL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:89_4"/><i><font color="Green" title="E48">A26</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I'</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I'</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:89_4"/><i><font color="Green" title="E49">A27</font></i>: 
( <font color="Maroon" title="c9">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp;  ex <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">y'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:89_4"/><i><font color="Green" title="E50">A28</font></i>: 
( <font color="Maroon" title="c10">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> &amp;  ex <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:89_4"><i><font color="Green" title="E48">A26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">lcx</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E7:89_4"/><i><font color="Green" title="E51">A29</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">lcx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:89_4"><i><font color="Green" title="E49">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">lcy</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E9:89_4"/><i><font color="Green" title="E52">A30</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c12">lcy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:89_4"><i><font color="Green" title="E50">A28</font></i></a></i>;<br/>
<a NAME="E10:89_4"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">lcx</font> <a href="ideal_1.html#K4" title="IDEAL_1:func.4">^</a> <font color="Maroon" title="c12">lcy</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y'</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E7:89_4"><i><font color="Green" title="E51">A29</font></i></a>, <a class="txt" href="ideal_1.html#E9:89_4"><i><font color="Green" title="E52">A30</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>;<br/>
<b>hence </b><a NAME="E11:89_4"/>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:89_4"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="ideal_1.html#E5:89_4"><i><font color="Green" title="E50">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:89"/><i><font color="Green" title="E48">A31</font></i>: 
<font color="Maroon" title="c6">I'</font> is <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">IDEAL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:89_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:89_5"/><i><font color="Green" title="E49">A32</font></i>: 
( <font color="Maroon" title="c9">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp;  ex <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">lcx</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E5:89_5"/><i><font color="Green" title="E50">A33</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c10">lcx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:89_5"><i><font color="Green" title="E49">A32</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">plcx</font> = <font color="Maroon" title="c7">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c10">lcx</font> as    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> <b>by </b><i><a class="ref" href="ideal_1.html#T23" target="_self" title="IDEAL_1:th.23">Th23</a></i>;<br/>
<a NAME="E7:89_5"/>
<font color="Maroon" title="c7">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">plcx</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E3:89_5"><i><font color="Green" title="E49">A32</font></i></a>, <a class="txt" href="ideal_1.html#E5:89_5"><i><font color="Green" title="E50">A33</font></i></a>, <a class="ref" href="binom.html#T4" title="BINOM:th.4">BINOM:4</a></i>;<br/>
<b>hence </b><a NAME="E8:89_5"/>
<font color="Maroon" title="c7">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:89"/>
<font color="Maroon" title="c6">I'</font> is <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D3" target="_self" title="IDEAL_1:def.3">IDEAL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:89_6"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:89_6"/><i><font color="Green" title="E49">A34</font></i>: 
( <font color="Maroon" title="c9">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp;  ex <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">lcx</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E5:89_6"/><i><font color="Green" title="E50">A35</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c10">lcx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:89_6"><i><font color="Green" title="E49">A34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">lcxp</font> = <font color="Maroon" title="c10">lcx</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c7">p</font> as    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> <b>by </b><i><a class="ref" href="ideal_1.html#T24" target="_self" title="IDEAL_1:th.24">Th24</a></i>;<br/>
<a NAME="E7:89_6"/>
<font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">lcxp</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E3:89_6"><i><font color="Green" title="E49">A34</font></i></a>, <a class="txt" href="ideal_1.html#E5:89_6"><i><font color="Green" title="E50">A35</font></i></a>, <a class="ref" href="binom.html#T5" title="BINOM:th.5">BINOM:5</a></i>;<br/>
<b>hence </b><a NAME="E8:89_6"/>
<font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:89"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E2:89"><i><font color="Green" title="E44">A19</font></i></a>, <a class="txt" href="ideal_1.html#E8:89"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="ideal_1.html#E9:89"><i><font color="Green" title="E48">A31</font></i></a>, <a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def15</a></i>;<br/>
<a NAME="E12:89"/><b>then </b><i><font color="Green" title="E49">A36</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<b>by </b><i><a class="txt" href="ideal_1.html#E1:89"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:89_7"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c8">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:89_7"/><i><font color="Green" title="E50">A37</font></i>: 
( <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp;  ex <font color="Olive" title="b1">lc</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E12:89"><i><font color="Green" title="E49">A36</font></i></a></i>;<br/><b>thus </b><a NAME="E4:89_7"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">f</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:89_7"><i><font color="Green" title="E50">A37</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E14:89"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">f</font>
 ;<br/>

<b>hence </b><a NAME="E15:89"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E12:89"><i><font color="Green" title="E49">A36</font></i></a></i>;<br/>


</div>
