<?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="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 ;<br/>

<b>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> </span>}</span> ;<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  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">N</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ex <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  &amp; not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Olive">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Olive">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">a<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">a<sub>2</sub></font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>] );<br/>
<a NAME="E2:54"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">n</font>, <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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</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="54_1"><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  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_2"/>
( <font color="Maroon">N</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">N</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="54_1_2_1"><b>suppose </b></a><a NAME="E1:54_1_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">N</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>assume </b><a NAME="E2:54_1_2_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>then consider </b><font color="Maroon">ww</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54_1_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">ww</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 <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">www</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:54_1_2_1"/>
<font color="Maroon">www</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font>
 <b>and </b><br/><a NAME="E7:54_1_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">www</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">www</font> )
 <b>by </b><i/>;<br/><a NAME="E8:54_1_2_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">www</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:54_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,1]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_2"><b>suppose </b></a><a NAME="E1:54_1_2_2"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">N</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon">B</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>consider </b><font color="Maroon">BD1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_1_2_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">BD1</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:54_1_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">BD1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="txt" href="dickson.html#E1:54"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E5:54_1_2_2"/>
<font color="Maroon">BD1</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:54_1_2_2"/><b>then </b>
<font color="Maroon">BD1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E7:54_1_2_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">BBB</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E9:54_1_2_2"/><i><font color="Green">E69</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">BBB</font>
 <b>and </b><br/><a NAME="E10:54_1_2_2"/>
<font color="Maroon">BBB</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E11:54_1_2_2"/><i><font color="Green">E70</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E12:54_1_2_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</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/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">x</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">x</font>;<br/><b>take </b>
<font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">x</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">x</font>
;<br/><b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> ;<br/><b>assume </b><a NAME="E14:54_1_2_2"/><i><font color="Green">E71</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>take </b>
<font color="Maroon">N</font>
;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/><b>take </b>
<font color="Maroon">B</font>
;<br/><b>thus </b><a NAME="E16:54_1_2_2"/>
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 ;<br/><b>thus </b><a NAME="E17:54_1_2_2"/>
not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a></i>;<br/><b>thus </b><a NAME="E18:54_1_2_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 ;<br/><b>thus </b><a NAME="E19:54_1_2_2"/>
<font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">x</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">x</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>,<font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="txt" href="dickson.html#E1:54"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">BALL</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">b1</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<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="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:54"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="dickson.html#K7" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>
 <b>and </b><br/><a NAME="E9:54"/><i><font color="Green">E75</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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</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>();<br/></i>
<a NAME="E10:54"/><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:54"/><i><font color="Green">E90</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><a NAME="E1:54_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:54_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>set </b><font color="Maroon">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> ;<br/><a NAME="E3:54_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">b</font>)
 <b>and </b><br/><a NAME="E4:54_2_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_2_1"/>
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_2"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:54_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:54_2_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E5:54_2_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">sx</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E8:54_2_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E9:54_2_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">sx</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E10:54_2_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/><a NAME="E12:54_2_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">b</font>)
 <b>by </b><i/>;<br/><a NAME="E13:54_2_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">b</font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E14:54_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E7:54_2"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E8:54_2"/>
contradiction
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">tb</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E14:54"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">tb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">tb</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<a NAME="E15:54"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">tb</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E16:54"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">bt</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E18:54"/><i><font color="Green">E125</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">bt</font>
 <b>and </b><br/><a NAME="E19:54"/><i><font color="Green">E126</font></i>: 
 ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bt</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<b>consider </b><font color="Maroon">b'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E21:54"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bt</font>
 <b>and </b><br/><a NAME="E22:54"/>
not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">b'</font>) is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E23:54"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/>
<a NAME="E24:54"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, <a class="txt" href="dickson.html#E2:54"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E25:54"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">eb</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E27:54"/><i><font color="Green">E129</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b'</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">eb</font>
 <b>and </b><br/><a NAME="E28:54"/><i><font color="Green">E130</font></i>: 
 ex <font color="Olive">eb'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">eb'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">eb</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">eb'</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<b>consider </b><font color="Maroon">eb'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E30:54"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">eb'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">eb</font>
 <b>and </b><br/><a NAME="E31:54"/><i><font color="Green">E132</font></i>: 
not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">eb'</font>) is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<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">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">W3</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span> ;<br/>
<a NAME="E32:54"/><i><font color="Green">E133</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="finseq_1.html#S6">FINSEQ_1:sch 6</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_3_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:54_3_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 ;<br/><a NAME="E5:54_3_1"/><i><font color="Green">E136</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E6:54_3_1"/>
( <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 or <font color="Maroon">w</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 )
 ;<br/><a NAME="E7:54_3_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span>  )
 <b>by </b><i>, , <a class="ref" href="dickson.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E8:54_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_3"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2_1"><b>suppose </b></a><a NAME="E1:54_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_3_2_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2_2"><b>suppose </b></a><a NAME="E1:54_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_3_2_2"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E34:54"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">w</font>] ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E35:54"/><b>then </b><i><font color="Green">E140</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, , , , , <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>];<br/>
<a NAME="E36:54"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:54_4"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>consider </b><font color="Maroon">N</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:54_4"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:54_4"/><i><font color="Green">E169</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E5:54_4"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E6:54_4"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">BALL</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon">iN</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">BD</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:54_4"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">BD</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> &amp; <font color="Maroon">BD</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, <a class="txt" href="dickson.html#E1:54"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:54_4"/>
<font color="Maroon">BD</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#D3" target="_self">Def3</a>, <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E10:54_4"/><b>then </b>
<font color="Maroon">BD</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E11:54_4"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">BB</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E13:54_4"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">BB</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E14:54_4"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#D4" target="_self">Def4</a>, <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1"><b>now </b></a><div class="add"><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>);<br/><a NAME="E1:54_4_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="dickson.html#D4" target="_self">Def4</a></i>;<br/><b>assume </b><a NAME="E2:54_4_1"/><i><font color="Green">E176</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>set </b><font color="Maroon">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> ;<br/><a NAME="E3:54_4_1"/><i><font color="Green">E177</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_4_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E3:54_4_1_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>)
 <b>and </b><br/><a NAME="E4:54_4_1_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_4_1_1"/>
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="txt" href="dickson.html#E4:54"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_4_1"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_4_1"/>
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_4_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:54_4_1_2"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E5:54_4_1_2"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 ;<br/>
<a NAME="E6:54_4_1_2"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="txt" href="dickson.html#E5:54"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">sw</font> = <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E9:54_4_1_2"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E10:54_4_1_2"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">sw</font>
 <b>by </b><i>, <a class="txt" href="dickson.html#E9:54"><i><font color="Green">E75</font></i></a>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E11:54_4_1_2"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E12:54_4_1_2"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_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="E1:54_4_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_1_2_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E4:54_4_1_2_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E5:54_4_1_2_1"/>
<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:54_4_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_4_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_4_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <b>by </b><i><a class="ref" href="dickson.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E15:54_4_1_2"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>)
 
<b>by </b><i><a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a>, <a class="ref" href="dickson.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E16:54_4_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="dickson.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E17:54_4_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green">E72</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:54_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="dickson.html#T8" target="_self">Th8</a>, , <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E17:54_4"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E18:54_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E19:54_4"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E21:54_4"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>and </b><br/><a NAME="E22:54_4"/><i><font color="Green">E231</font></i>: 
 ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><b>consider </b><font color="Maroon">b'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E24:54_4"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E25:54_4"/><i><font color="Green">E233</font></i>: 
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/><a NAME="E26:54_4"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E27:54_4"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E28:54_4"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:54"><i><font color="Green">E73</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E30:54_4"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E31:54_4"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E32:54_4"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 <b>by </b><i/>;<br/><a NAME="E33:54_4"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i>, , <a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E9:54"><i><font color="Green">E75</font></i></a>, </i>;<br/><a NAME="E34:54_4"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E9:54"><i><font color="Green">E75</font></i></a>, </i>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">W1</font> = <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">W2</font> = <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">W3</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span> ;<br/><a NAME="E35:54_4"/><i><font color="Green">E240</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="dickson.html#S1" target="_self">DICKSON:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_2_1"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:54_4_2_1"/><i><font color="Green">E242</font></i>: 
( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> )
 ;<br/><a NAME="E5:54_4_2_1"/>
( <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">w</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><a NAME="E6:54_4_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:54_4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_4_2"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_4_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2_1"><b>suppose </b></a><a NAME="E1:54_4_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_2_2_1"/><i><font color="Green">E244</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> )
 ;<br/><a NAME="E4:54_4_2_2_1"/>
<font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="dickson.html#D5" target="_self">Def5</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:54_4_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2_2"><b>suppose </b></a><a NAME="E1:54_4_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_2_2_2"/><i><font color="Green">E245</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_4_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E37:54_4"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">w1</font>) where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> &amp; <font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">w1</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E38:54_4"/><b>then </b>
not <span class="p1">{<span class="default"> <font color="Olive">w1</font> where <font color="Olive">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w1</font> &amp; <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w1</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="txt" href="dickson.html#E4:54"><i><font color="Green">E72</font></i></a>, <a class="ref" href="dickson.html#T9" target="_self">Th9</a>, , <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/><b>hence </b><a NAME="E39:54_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E38:54"/><i><font color="Green">E246</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>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="dickson.html#T7" target="_self">Th7</a>);<br/></i>
<b>set </b><font color="Maroon">t</font> = <font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>;<br/>
<b>take </b>
<font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1"><b>now </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/><a NAME="E1:54_5_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a></i>;<br/><b>consider </b><font color="Maroon">N</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:54_5_1"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">fn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E5:54_5_1"/><i><font color="Green">E248</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">fn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E6:54_5_1"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E7:54_5_1"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">fn</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon">BJ</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">BC</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">BD</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:54_5_1"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">BD</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> &amp; <font color="Maroon">BD</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, <a class="txt" href="dickson.html#E1:54"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E10:54_5_1"/>
<font color="Maroon">BD</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#D3" target="_self">Def3</a>, <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E11:54_5_1"/><b>then </b>
<font color="Maroon">BD</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E12:54_5_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">BB</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E14:54_5_1"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BB</font>
 <b>and </b><br/><a NAME="E15:54_5_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">BB</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E16:54_5_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#D4" target="_self">Def4</a>, <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E17:54_5_1"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1"><b>now </b></a><div class="add"><a NAME="E1:54_5_1_1"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="dickson.html#D4" target="_self">Def4</a>, </i>;<br/><b>assume </b><a NAME="E2:54_5_1_1"/><i><font color="Green">E259</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">fn</font>);<br/><b>set </b><font color="Maroon">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> ;<br/><b>set </b><font color="Maroon">iN</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">fn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> ;<br/><a NAME="E3:54_5_1_1"/><i><font color="Green">E260</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_5_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E3:54_5_1_1_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">fn</font>)
 <b>and </b><br/><a NAME="E4:54_5_1_1_1"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_5_1_1_1"/>
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="txt" href="dickson.html#E5:54"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_5_1_1"/><b>then </b><i><font color="Green">E263</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_5_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">fn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_5_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">fn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_5_1_1_2"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:54_5_1_1_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E5:54_5_1_1_2"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 ;<br/>
<a NAME="E6:54_5_1_1_2"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="txt" href="dickson.html#E9:54"><i><font color="Green">E75</font></i></a>, <a class="ref" href="dickson.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sw</font> = <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E9:54_5_1_1_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E10:54_5_1_1_2"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">sw</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#D5" target="_self">Def5</a>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E11:54_5_1_1_2"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E12:54_5_1_1_2"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_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="E1:54_5_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_5_1_1_2_1"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E4:54_5_1_1_2_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E5:54_5_1_1_2_1"/>
<font color="Maroon">fn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:54_5_1_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_5_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_5_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/>
<a NAME="E15:54_5_1_1_2"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">fn</font>)
 
<b>by </b><i><a class="ref" href="dickson.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E16:54_5_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">fn</font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:54_5_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:54_5_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="dickson.html#T8" target="_self">Th8</a>, <a class="txt" href="dickson.html#E5:54"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E20:54_5_1"/><i><font color="Green">E273</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E21:54_5_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E22:54_5_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E24:54_5_1"/><i><font color="Green">E274</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E25:54_5_1"/>
 ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E26:54_5_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E28:54_5_1"/><i><font color="Green">E275</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">fn</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">fn</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">fn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E29:54_5_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="dickson.html#E4:54"><i><font color="Green">E72</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:54_5"/>
<font color="Maroon">f</font> is <a href="seqm_3.html#V1">increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/><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/><a NAME="E3:54_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:54_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i><a class="ref" href="seqm_3.html#T29">SEQM_3:29</a></i>;<br/>
<a NAME="E42:54"/>
<font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E43:54"/>
<font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>

<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D2" target="_self">DICKSON:def 2</a><br/>

<a NAME="E44:54"/><i><font color="Green">E276</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E45:54"/><i><font color="Green">E277</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">N</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E47:54"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E48:54"/><i><font color="Green">E279</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E49:54"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E50:54"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">BX</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">sfn1</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">BD</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E52:54"/><i><font color="Green">E284</font></i>: 
( <font color="Maroon">BD</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> &amp; <font color="Maroon">BD</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, <a class="txt" href="dickson.html#E1:54"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E53:54"/>
<font color="Maroon">BD</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E54:54"/><b>then </b>
<font color="Maroon">BD</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E55:54"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">BB</font> where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font> : <font color="Olive">BB</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon">BB</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">N</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E57:54"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">BB</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E58:54"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">BB</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>
<a NAME="E59:54"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6"><b>now </b></a><div class="add"><a NAME="E1:54_6"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, </i>;<br/><b>assume </b><a NAME="E2:54_6"/><i><font color="Green">E289</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>);<br/><b>set </b><font color="Maroon">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> ;<br/><b>set </b><font color="Maroon">iN</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> ;<br/><a NAME="E3:54_6"/><i><font color="Green">E290</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_6_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E3:54_6_1"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>)
 <b>and </b><br/><a NAME="E4:54_6_1"/><i><font color="Green">E292</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_6_1"/>
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E9:54"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_6"/><b>then </b><i><font color="Green">E293</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E5:54"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_6"/>
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">w</font> &amp; <font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">w</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_6_2"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:54_6_2"/><i><font color="Green">E295</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E5:54_6_2"/><i><font color="Green">E296</font></i>: 
<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 ;<br/>
<a NAME="E6:54_6_2"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T9" target="_self">Th9</a>, <a class="ref" href="dickson.html#D5" target="_self">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sw</font> = <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E9:54_6_2"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E10:54_6_2"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">sw</font>
 <b>by </b><i>, , <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E11:54_6_2"/><i><font color="Green">E300</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E12:54_6_2"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_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="E1:54_6_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_6_2_1"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E4:54_6_2_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E5:54_6_2_1"/>
<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:54_6_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_6_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_6_2"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/>
<a NAME="E15:54_6_2"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>)
 
<b>by </b><i><a class="ref" href="dickson.html#D5" target="_self">Def5</a>, </i>;<br/>
<a NAME="E16:54_6_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:54_6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E9:54"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:54_6"/>
contradiction
 <b>by </b><i>, <a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E62:54"/><i><font color="Green">E303</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<a NAME="E63:54"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E64:54"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E66:54"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>and </b><br/><a NAME="E67:54"/>
 ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<a NAME="E68:54"/><i><font color="Green">E305</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E69:54"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E70:54"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E5:54"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E72:54"/><i><font color="Green">E306</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> :  ex <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b'</font>,<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E73:54"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E74:54"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">f</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 <b>by </b><i/>;<br/>
<a NAME="E75:54"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T8" target="_self">Th8</a>, <a class="ref" href="dickson.html#D4" target="_self">Def4</a>, <a class="txt" href="dickson.html#E8:54"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E9:54"><i><font color="Green">E75</font></i></a>, <a class="ref" href="dickson.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E76:54"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div>
