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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a> <a href="binom.html#V1" title="BINOM:attr.1">add-cancelable</a> <a href="ideal_1.html#V7" title="IDEAL_1:attr.7">Noetherian</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">B</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">R</font>;<br/>



<a NAME="E1:165"/>
<font color="Maroon" title="c2">B</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">finitely_generated</a>
 
<b>by </b><i><a class="ref" href="ideal_1.html#D27" target="_self" title="IDEAL_1:def.27">Def27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">D</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:165"/><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</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#D26" target="_self" title="IDEAL_1:def.26">Def26</a></i>;<br/>
<a NAME="E4:165"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</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#E3:165"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def15</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">cL</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">B</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Olive" title="b1">cL</font> &amp;  ex <font color="Olive" title="b2">fsB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font> st <br/>(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">cL</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">fsB</font> &amp; ( for <font color="Olive" title="b3">i</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="Olive" title="b3">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> <font color="Olive" title="b1">cL</font> holds <br/> ex <font color="Olive" title="b4">u</font>, <font color="Olive" title="b5">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Olive" title="b1">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b3">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b5">v</font> ) ) );<br/>
<a NAME="E5:165"/><i><font color="Green" title="E68">A3</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="165_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:165_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">cL</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E3:165_1"/><i><font color="Green" title="E69">A4</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">cL</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="ideal_1.html#T60" target="_self" title="IDEAL_1:th.60">Th60</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="165_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:165_1_1"/>
(  not <font color="Maroon" title="c5">cL</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c5">cL</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="165_1_1_1"><b>suppose </b></a><a NAME="E1:165_1_1_1"/><i><font color="Green" title="E70">A5</font></i>: 
not <font color="Maroon" title="c5">cL</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font>] <b>means </b> 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">R</font> st <font color="Maroon" title="c5">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <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="group_1.html#K1" title="GROUP_1:func.1">*</a> $2</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">v</font>;<br/><div><i><font color="Green" title="E71">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="165_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="E1:165_1_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">cL</font></span>)</span>
 ;<br/><a NAME="E2:165_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">k</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> <font color="Maroon" title="c5">cL</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c9">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E4:165_1_1_1_1"/><i><font color="Green" title="E72">A7</font></i>: 
<font color="Maroon" title="c5">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">v</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def9</a></i>;<br/><b>take </b><font color="Maroon" title="c10">d</font> = <font color="Maroon" title="c9">a</font>;<br/><b>thus </b><a NAME="E5:165_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c10">d</font>]
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165_1_1_1_1"><i><font color="Green" title="E72">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c6">fsB</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E4:165_1_1_1"/><i><font color="Green" title="E72">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">cL</font></span>)</span>
 <b>and </b><br/><a NAME="E5:165_1_1_1"/><i><font color="Green" title="E73">A9</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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">cL</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c6">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="ideal_1.html#E2:165_1_1_1"><i><font color="Green" title="E71">A6</font></i></a>);<br/></i><b>take </b><font color="Maroon" title="c7">u</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">fsB</font>;<br/><b>thus </b><a NAME="E6:165_1_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">B</font>
 ;<br/><a NAME="E7:165_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">cL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">cL</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E8:165_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c7">u</font>]
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:165_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="ideal_1.html#E1:165_1_1_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="ideal_1.html#E4:165_1_1_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="txt" href="ideal_1.html#E5:165_1_1_1"><i><font color="Green" title="E73">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="165_1_1_2"><b>suppose </b></a><a NAME="E1:165_1_1_2"/><i><font color="Green" title="E70">A10</font></i>: 
<font color="Maroon" title="c5">cL</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c6">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font>;<br/><b>set </b><font color="Maroon" title="c7">kL</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></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="165_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:165_1_1_2_1"/>
<font color="Maroon" title="c8">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><a NAME="E2:165_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:165_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E4:165_1_1_2_1"/><b>then </b><i><font color="Green" title="E71">A11</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c9">u</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>;<br/><b>take </b><font color="Maroon" title="c10">v</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>;<br/><b>take </b><font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c6">b</font>;<br/><b>thus </b><a NAME="E5:165_1_1_2_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <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="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165_1_1_2_1"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">kL</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">B</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="E5:165_1_1_2"/>
<font color="Maroon" title="c5">cL</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">R</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_1_1_2"><i><font color="Green" title="E70">A10</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E71">A12</font></i>: <font color="Maroon" title="c4">e</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="ideal_1.html#E3:165_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#T2" title="BINOM:th.2">BINOM:2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">kL</font>
<b>by </b><i><a class="ref" href="binom.html#T3" title="BINOM:th.3">BINOM:3</a></i>
;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font>] <b>means </b> 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">R</font> st <font color="Maroon" title="c8">kL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <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="group_1.html#K1" title="GROUP_1:func.1">*</a> $2</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">v</font>;<br/><div><i><font color="Green" title="E72">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="165_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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="E1:165_1_1_2_3"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">kL</font></span>)</span>
 ;<br/><a NAME="E2:165_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E3:165_1_1_2_3"/><b>then </b><i><font color="Green" title="E73">A14</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c6">b</font>;<br/><b>thus </b><a NAME="E4:165_1_1_2_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="165_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c11">u</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>;<br/>

<b>take </b><font color="Maroon" title="c12">v</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>;<br/>

<b>thus </b><a NAME="E1:165_1_1_2_3_1"/>
<font color="Maroon" title="c8">kL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:165_1_1_2_3"><i><font color="Green" title="E73">A14</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c9">fsB</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E9:165_1_1_2"/><i><font color="Green" title="E73">A15</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">kL</font></span>)</span>
 <b>and </b><br/><a NAME="E10:165_1_1_2"/><i><font color="Green" title="E74">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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">kL</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c9">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="ideal_1.html#E7:165_1_1_2"><i><font color="Green" title="E72">A13</font></i></a>);<br/></i><b>take </b><font color="Maroon" title="c10">u</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">fsB</font>;<br/><b>thus </b><a NAME="E11:165_1_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">B</font>
 ;<br/><a NAME="E12:165_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">kL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">kL</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E13:165_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c10">u</font>]
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:165_1_1_2"><i><font color="Green" title="E71">A12</font></i></a>, <a class="txt" href="ideal_1.html#E9:165_1_1_2"><i><font color="Green" title="E73">A15</font></i></a>, <a class="txt" href="ideal_1.html#E10:165_1_1_2"><i><font color="Green" title="E74">A16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">D</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E7:165"/><i><font color="Green" title="E69">A17</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="ideal_1.html#E5:165"><i><font color="Green" title="E68">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">rf</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c2">B</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">C</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">rf</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">B</font> ;<br/>
<b>consider </b><font color="Maroon" title="c7">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:165"/><i><font color="Green" title="E70">A18</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">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>consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:165"/><i><font color="Green" title="E71">A19</font></i>: 
( <font color="Maroon" title="c8">x</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> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E11:165"><i><font color="Green" title="E70">A18</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:165"/><i><font color="Green" title="E72">A20</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">cL</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E16:165"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c9">cL</font>
 <b>and </b><br/><a NAME="E17:165"/><i><font color="Green" title="E73">A21</font></i>: 
 ex <font color="Olive" title="b1">fsB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font> st <br/>(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">cL</font> &amp; <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">fsB</font> &amp; ( for <font color="Olive" title="b2">i</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="Olive" title="b2">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> <font color="Maroon" title="c9">cL</font> holds <br/> ex <font color="Olive" title="b3">u</font>, <font color="Olive" title="b4">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c9">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b4">v</font> ) )
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165"><i><font color="Green" title="E69">A17</font></i></a>, <a class="txt" href="ideal_1.html#E13:165"><i><font color="Green" title="E71">A19</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">fsB</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E19:165"/><i><font color="Green" title="E74">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">cL</font>
 <b>and </b><br/><a NAME="E20:165"/><i><font color="Green" title="E75">A23</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">fsB</font>
 <b>and </b><br/><a NAME="E21:165"/>
for <font color="Olive" title="b1">i</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="Olive" title="b1">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> <font color="Maroon" title="c9">cL</font> holds <br/> ex <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c9">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E17:165"><i><font color="Green" title="E73">A21</font></i></a></i>;<br/>
<a NAME="E22:165"/>
not <font color="Maroon" title="c7">r</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E19:165"><i><font color="Green" title="E74">A22</font></i></a>, <a class="txt" href="ideal_1.html#E20:165"><i><font color="Green" title="E75">A23</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:165"/><i><font color="Green" title="E76">A24</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">r</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/>
<a NAME="E25:165"/><i><font color="Green" title="E77">A25</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="165_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">r</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:165_2"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:165_2"/><i><font color="Green" title="E78">A26</font></i>: 
( <font color="Maroon" title="c13">x</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> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">cL</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E5:165_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c14">cL</font>
 <b>and </b><br/><a NAME="E6:165_2"/><i><font color="Green" title="E79">A27</font></i>: 
 ex <font color="Olive" title="b1">fsB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font> st <br/>(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">cL</font> &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">fsB</font> &amp; ( for <font color="Olive" title="b2">i</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="Olive" title="b2">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> <font color="Maroon" title="c14">cL</font> holds <br/> ex <font color="Olive" title="b3">u</font>, <font color="Olive" title="b4">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c14">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b4">v</font> ) )
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165"><i><font color="Green" title="E69">A17</font></i></a>, <a class="txt" href="ideal_1.html#E3:165_2"><i><font color="Green" title="E78">A26</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">fsB</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E8:165_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">cL</font>
 <b>and </b><br/><a NAME="E9:165_2"/><i><font color="Green" title="E80">A28</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">fsB</font>
 <b>and </b><br/><a NAME="E10:165_2"/>
for <font color="Olive" title="b1">i</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="Olive" title="b1">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> <font color="Maroon" title="c14">cL</font> holds <br/> ex <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c14">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:165_2"><i><font color="Green" title="E79">A27</font></i></a></i>;<br/><b>thus </b><a NAME="E11:165_2"/>
<font color="Maroon" title="c12">r</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="ideal_1.html#E9:165_2"><i><font color="Green" title="E80">A28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">C</font> = <font color="Maroon" title="c6">C</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="ideal_1.html#E11:165"><i><font color="Green" title="E70">A18</font></i></a>, <a class="txt" href="ideal_1.html#E24:165"><i><font color="Green" title="E76">A24</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="165_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:165_3"/><i><font color="Green" title="E78">A29</font></i>: 
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ;<br/><a NAME="E2:165_3"/><b>then </b>
<font color="Maroon" title="c13">d</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> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:165_3"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:165_3"/><b>then </b><i><font color="Green" title="E79">A30</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">C</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">cL</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E6:165_3"/><i><font color="Green" title="E80">A31</font></i>: 
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c14">cL</font>
 <b>and </b><br/><a NAME="E7:165_3"/><i><font color="Green" title="E81">A32</font></i>: 
 ex <font color="Olive" title="b1">fsB</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font> st <br/>(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">cL</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">fsB</font> &amp; ( for <font color="Olive" title="b2">i</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="Olive" title="b2">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> <font color="Maroon" title="c14">cL</font> holds <br/> ex <font color="Olive" title="b3">u</font>, <font color="Olive" title="b4">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c14">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b4">v</font> ) )
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165"><i><font color="Green" title="E69">A17</font></i></a>, <a class="txt" href="ideal_1.html#E1:165_3"><i><font color="Green" title="E78">A29</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="165_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:165_3_1"/><i><font color="Green" title="E82">A33</font></i>: 
<font color="Maroon" title="c15">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> <font color="Maroon" title="c14">cL</font>
 ;<br/><b>consider </b><font color="Maroon" title="c16">fsB</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E3:165_3_1"/><i><font color="Green" title="E83">A34</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">fsB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">cL</font>
 <b>and </b><br/><a NAME="E4:165_3_1"/><i><font color="Green" title="E84">A35</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">fsB</font>
 <b>and </b><br/><a NAME="E5:165_3_1"/><i><font color="Green" title="E85">A36</font></i>: 
for <font color="Olive" title="b1">i</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="Olive" title="b1">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> <font color="Maroon" title="c14">cL</font> holds <br/> ex <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c14">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165_3"><i><font color="Green" title="E81">A32</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">u</font>, <font color="Maroon" title="c18">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E7:165_3_1"/><i><font color="Green" title="E86">A37</font></i>: 
<font color="Maroon" title="c14">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_3_1"><i><font color="Green" title="E82">A33</font></i></a>, <a class="txt" href="ideal_1.html#E5:165_3_1"><i><font color="Green" title="E85">A36</font></i></a></i>;<br/><a NAME="E8:165_3_1"/>
<font color="Maroon" title="c16">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fsB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_3_1"><i><font color="Green" title="E82">A33</font></i></a>, <a class="txt" href="ideal_1.html#E3:165_3_1"><i><font color="Green" title="E83">A34</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E9:165_3_1"/><b>then </b>
<font color="Maroon" title="c16">fsB</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">d</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_3_1"><i><font color="Green" title="E82">A33</font></i></a>, <a class="txt" href="ideal_1.html#E3:165_3_1"><i><font color="Green" title="E83">A34</font></i></a>, <a class="txt" href="ideal_1.html#E4:165_3_1"><i><font color="Green" title="E84">A35</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:165_3_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">R</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="c12">C</font> st <font color="Maroon" title="c14">cL</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">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="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165_3"><i><font color="Green" title="E79">A30</font></i></a>, <a class="txt" href="ideal_1.html#E7:165_3_1"><i><font color="Green" title="E86">A37</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">cL'</font> = <font color="Maroon" title="c14">cL</font> as    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c12">C</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="E10:165_3"/>
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c15">cL'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:165_3"><i><font color="Green" title="E80">A31</font></i></a></i>;<br/><b>hence </b><a NAME="E11:165_3"/>
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</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#T60" target="_self" title="IDEAL_1:th.60">Th60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E29:165"/><b>then </b>
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">C</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="E30:165"/><b>then </b>
<font color="Maroon" title="c3">D</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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">C</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#T57" target="_self" title="IDEAL_1:th.57">Th57</a></i>;<br/>
<a NAME="E31:165"/><b>then </b><i><font color="Green" title="E78">A38</font></i>: 
<font color="Maroon" title="c2">B</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="c12">C</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#E3:165"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="ideal_1.html#T44" target="_self" title="IDEAL_1:th.44">Th44</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">C</font>
;<br/>

<a NAME="E32:165"/>
<font color="Maroon" title="c12">C</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="c2">B</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#T57" target="_self" title="IDEAL_1:th.57">Th57</a></i>;<br/>
<b>hence </b><a NAME="E33:165"/>
( <font color="Maroon" title="c12">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c12">C</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</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#E31:165"><i><font color="Green" title="E78">A38</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
