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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:169"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>  holds <br/>(  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is  not   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> or  ex <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; not <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> ) )
 <b>and </b><br/><a NAME="E2:169"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">R</font> is <a href="ideal_1.html#V7" target="_self">Noetherian</a>
 ;<br/>

<b>consider </b><font color="Maroon">I</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:169"/><i><font color="Green">E43</font></i>: 
not <font color="Maroon">I</font> is <a href="ideal_1.html#V6" target="_self">finitely_generated</a>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:169"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : <font color="Olive">S</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> </span>}</span> ;<br/>
<a NAME="E8:169"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">e</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:169"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">e</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : <font color="Olive">S</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> </span>}</span> 
 
;<br/>
<a NAME="E10:169"/>
<span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : <font color="Olive">S</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:169_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : <font color="Olive">S</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> </span>}</span> 
 ;<br/>

<a NAME="E2:169_1"/><b>then </b>
 ex <font color="Olive">s</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> )
 
;<br/>
<b>hence </b><a NAME="E3:169_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : <font color="Olive">S</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">e'</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">e</font></span><a href="domain_1.html#K6">}</a></span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">r</font></span><a href="domain_1.html#K6">}</a></span> );<br/>
<a NAME="E13:169"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_2"><b>proof </b></a><div class="add">

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



<a NAME="E1:169_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:169_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:169_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x'</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 ;<br/>
<a NAME="E5:169_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:169_2"/>
<font color="Maroon">x'</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i>, <a class="txt" href="ideal_1.html#E2:169"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E7:169_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:169_2"/><b>then </b>
not <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:169_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</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/>
<b>reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">x1'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E12:169_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x1'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:169_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:169_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1'</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:169_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E13:169"><i><font color="Green">E76</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:169_2"/><b>then </b>
<font color="Maroon">x1'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:169_2"/><b>then </b>
<font color="Maroon">x1'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E3:169_2"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x1'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="txt" href="ideal_1.html#E13:169"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E17:169_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E27"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="ideal_1.html#E13:169"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">D</font><b> such that </b><br/><a NAME="E15:169"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="ideal_1.html#E9:169"><i><font color="Green">E72</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>] <b>means </b> ex <font color="Olive">c</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  );<br/>
<a NAME="E16:169"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_3"><b>proof </b></a><div class="add">

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

<a NAME="E1:169_3"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:169_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">c</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">c</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

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

<b>thus </b><a NAME="E5:169_3"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E18:169"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E19:169"/><i><font color="Green">E123</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_4"><b>proof </b></a><div class="add">

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

<a NAME="E1:169_4"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:169_4"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E20:169"/>
for <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_5"><b>proof </b></a><div class="add">

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

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:169_5"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_5_1"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:169_5_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:169_5_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i/>;<br/>
<a NAME="E4:169_5_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:169_5_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E7:169_5_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c'</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 ;<br/>
<b>consider </b><font color="Maroon">c1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E9:169_5_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E10:169_5_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i/>;<br/>
<a NAME="E11:169_5_1"/>
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">c1'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E13:169_5_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">c1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 <b>and </b><br/><a NAME="E14:169_5_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">c1'</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 ;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E16:169_5_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span>
 <b>and </b><br/><a NAME="E17:169_5_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E27"><i><font color="Green">Lemma73</font></i></a>, , </i>;<br/>
<a NAME="E18:169_5_1"/>
<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#D10" target="_self">Def10</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E19:169_5_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T19" target="_self">Th19</a>, , , , <a class="ref" href="ideal_1.html#D9" target="_self">Def9</a>, <a class="txt" href="ideal_1.html#E2:169"><i><font color="Green">E42</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a><br/>

<a NAME="E20:169_5_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">c1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c1</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 
<b>by </b><i>, <a class="ref" href="ideal_1.html#D9" target="_self">Def9</a>, </i>;<br/>
<a NAME="E21:169_5_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E22:169_5_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#D10" target="_self">Def10</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E23:169_5_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:169_5"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
;<br/>
<a NAME="E3:169_5"/><i><font color="Green">E139</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_5_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:169_5_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E2:169_5_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a><br/>

<a NAME="E3:169_5_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:169_5_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:169_5_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a><br/>

<a NAME="E6:169_5_2"/>
not <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T19" target="_self">Th19</a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:169_5_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E9:169_5_2"/><i><font color="Green">E144</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><a NAME="E10:169_5_2"/>
not <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:169_5"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>assume </b><a NAME="E5:169_5"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E6:169_5"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:169_5"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E9:169_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:169_5"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T19" target="_self">Th19</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:169"/>
contradiction
 <b>by </b><i>, <a class="txt" href="ideal_1.html#E13:169"><i><font color="Green">E76</font></i></a></i>;<br/>


</div>
