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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1">add-cancelable</a> <a href="ideal_1.html#V7" target="_self">Noetherian</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:165"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  is <a href="ideal_1.html#V6" target="_self">finitely_generated</a>
 
<b>by </b><i><a class="ref" href="ideal_1.html#D27" target="_self">Def27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:165"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E4:165"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i><a class="txt" href="ideal_1.html#E3:165"><i><font color="Green">E67</font></i></a>, <a class="ref" href="ideal_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>5</sub></font> ) ) );<br/>
<a NAME="E5:165"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:165_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:165_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165"><i><font color="Green">E68</font></i></a>, <a class="ref" href="ideal_1.html#T60" target="_self">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">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">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">E71</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font>;<br/><div><i><font color="Green">E72</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">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:165_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><a NAME="E2:165_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:165_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self">Def9</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>thus </b><a NAME="E5:165_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165_1_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:165_1_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:165_1_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>(<a class="txt" href="ideal_1.html#E2:165_1_1_1"><i><font color="Green">E72</font></i></a>);<br/></i><b>take </b><font color="Maroon">c<sub>7</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E6:165_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E7:165_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">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">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:165_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="ideal_1.html#E1:165_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="ideal_1.html#E4:165_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="ideal_1.html#E5:165_1_1_1"><i><font color="Green">E74</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">E71</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&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="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&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">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:165_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:165_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&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="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:165_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E4:165_1_1_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E5:165_1_1_2_1"/>
<span class="p1"><a href="finseq_1.html#K12">&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="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165_1_1_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&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="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E5:165_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><b>then </b><i><font color="Green">E72</font></i>: <font color="Maroon">c<sub>4</sub></font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="ideal_1.html#E3:165_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#T2">BINOM:2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="binom.html#T3">BINOM:3</a></i>
;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font>;<br/><div><i><font color="Green">E73</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">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:165_1_1_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><a NAME="E2:165_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E3:165_1_1_2_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E4:165_1_1_2_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>11</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>12</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E1:165_1_1_2_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:165_1_1_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:165_1_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:165_1_1_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>(<a class="txt" href="ideal_1.html#E7:165_1_1_2"><i><font color="Green">E73</font></i></a>);<br/></i><b>take </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>;<br/><b>thus </b><a NAME="E11:165_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E12:165_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">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">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:165_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="ideal_1.html#E9:165_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="ideal_1.html#E10:165_1_1_2"><i><font color="Green">E75</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">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:165"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="ideal_1.html#E5:165"><i><font color="Green">E69</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:165"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:165"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E11:165"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:165"/><i><font color="Green">E73</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:165"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E17:165"/><i><font color="Green">E74</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165"><i><font color="Green">E70</font></i></a>, <a class="txt" href="ideal_1.html#E13:165"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E19:165"/><i><font color="Green">E75</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E20:165"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E21:165"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E17:165"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E22:165"/>
not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E19:165"><i><font color="Green">E75</font></i></a>, <a class="txt" href="ideal_1.html#E20:165"><i><font color="Green">E76</font></i></a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:165"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E25:165"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E14:165"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<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">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:165_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:165_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:165_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E6:165_2"/><i><font color="Green">E80</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165"><i><font color="Green">E70</font></i></a>, <a class="txt" href="ideal_1.html#E3:165_2"><i><font color="Green">E79</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:165_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E9:165_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E10:165_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:165_2"><i><font color="Green">E80</font></i></a></i>;<br/><b>thus </b><a NAME="E11:165_2"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="ideal_1.html#E9:165_2"><i><font color="Green">E81</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="ideal_1.html#E11:165"><i><font color="Green">E71</font></i></a>, <a class="txt" href="ideal_1.html#E24:165"><i><font color="Green">E77</font></i></a>, <a class="txt" href="ideal_1.html#E25:165"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, <a class="ref" href="xboole_1.html#T1">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">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:165_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:165_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:165_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:165_3"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:165_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E7:165_3"/><i><font color="Green">E82</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relset_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165"><i><font color="Green">E70</font></i></a>, <a class="txt" href="ideal_1.html#E1:165_3"><i><font color="Green">E79</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">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:165_3_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:165_3_1"/><i><font color="Green">E84</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:165_3_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:165_3_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:165_3"><i><font color="Green">E82</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:165_3_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_3_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="ideal_1.html#E5:165_3_1"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E8:165_3_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_3_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="ideal_1.html#E3:165_3_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E9:165_3_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:165_3_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="ideal_1.html#E3:165_3_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="ideal_1.html#E4:165_3_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:165_3_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font> st <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:165_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="ideal_1.html#E7:165_3_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E10:165_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:165_3"><i><font color="Green">E81</font></i></a></i>;<br/><b>hence </b><a NAME="E11:165_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T60" target="_self">Th60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E29:165"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E30:165"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E31:165"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i><a class="txt" href="ideal_1.html#E3:165"><i><font color="Green">E67</font></i></a>, <a class="ref" href="ideal_1.html#T44" target="_self">Th44</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<a NAME="E32:165"/>
<font color="Maroon">c<sub>12</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#T57" target="_self">Th57</a></i>;<br/>
<b>hence </b><a NAME="E33:165"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  )
 <b>by </b><i><a class="txt" href="ideal_1.html#E31:165"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
