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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">s</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( $1 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> &amp; $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : $1 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">N</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">B</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">w</font></span>)</span> where <font color="Olive" title="b3">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">w</font> &amp; $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">w</font> ) </span>}</span>  &amp; not <span class="p1">{<span class="default"> <font color="Olive" title="b3">w</font> where <font color="Olive" title="b3">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">w</font> &amp; $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">BB</font> where <font color="Olive" title="b3">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Olive" title="b1">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b3">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b3">b</font> where <font color="Olive" title="b3">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">B</font> :  ex <font color="Olive" title="b4">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b4">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">b'</font>,$2) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,$2 );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( not <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies <font color="Maroon">S<sub>1</sub></font>[$1,$2,$3] );<br/>
<a NAME="E2:54"/><i><font color="Green" title="E39">A2</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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" title="c4">n</font>, <font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>set </b><font color="Maroon" title="c6">N</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">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" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c8">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w</font> &amp; <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_1_1"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">N</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">W</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">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" title="c7">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c7">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_1"><b>suppose </b></a><a NAME="E1:54_1_2_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c7">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">ww</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:54_1_2_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c9">ww</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">www</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:54_1_2_1"/>
<font color="Maroon" title="c10">www</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ww</font>
 <b>and </b><br/><a NAME="E7:54_1_2_1"/><i><font color="Green" title="E42">A6</font></i>: 
( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">www</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">www</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_1_2_1"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/><a NAME="E8:54_1_2_1"/>
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">www</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:54_1_2_1"><i><font color="Green" title="E42">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E9:54_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,1]
 <b>by </b><i><a class="txt" href="dickson.html#E1:54_1_2_1"><i><font color="Green" title="E40">A4</font></i></a></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" title="E40">A7</font></i>: 
not <font color="Maroon" title="c7">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c9">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c10">B</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c11">BD1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_1_2_2"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c11">BD1</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E4:54_1_2_2"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c11">BD1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/><a NAME="E5:54_1_2_2"/>
<font color="Maroon" title="c11">BD1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E3:54_1_2_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></i>;<br/><a NAME="E6:54_1_2_2"/><b>then </b>
<font color="Maroon" title="c11">BD1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_1_2_2"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/><a NAME="E7:54_1_2_2"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c12">BBB</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E9:54_1_2_2"/><i><font color="Green" title="E43">A10</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">BBB</font>
 <b>and </b><br/><a NAME="E10:54_1_2_2"/>
<font color="Maroon" title="c12">BBB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><a NAME="E11:54_1_2_2"/><i><font color="Green" title="E44">A11</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E9:54_1_2_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></i>;<br/><a NAME="E12:54_1_2_2"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">B</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="dickson.html#E1:54_1_2_2"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="dickson.html#E11:54_1_2_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="dickson.html#T27" target="_self" title="DICKSON:th.27">Th27</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c5">x</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon" title="c5">x</font>;<br/><b>take </b>
<font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c5">x</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon" title="c5">x</font>
;<br/><b>set </b><font color="Maroon" title="c15">W</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> ;<br/><b>assume </b><a NAME="E14:54_1_2_2"/><i><font color="Green" title="E45">A12</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>take </b>
<font color="Maroon" title="c7">N</font>
;<br/><b>reconsider </b><font color="Maroon" title="c16">B</font> = <font color="Maroon" title="c13">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>take </b>
<font color="Maroon" title="c16">B</font>
;<br/><b>thus </b><a NAME="E16:54_1_2_2"/>
<font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">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" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_1_2_2"><i><font color="Green" title="E45">A12</font></i></a></i>;<br/><b>thus </b><a NAME="E18:54_1_2_2"/>
<font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 ;<br/><b>thus </b><a NAME="E19:54_1_2_2"/>
<font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c5">x</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c5">x</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon" title="c5">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:54"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="dickson.html#T27" target="_self" title="DICKSON:th.27">Th27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">BALL</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">b1</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:54"/><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>
 <b>and </b><br/><a NAME="E9:54"/><i><font color="Green" title="E43">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="dickson.html#E2:54"><i><font color="Green" title="E39">A2</font></i></a>);<br/></i>
<a NAME="E10:54"/><i><font color="Green" title="E44">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:54"/><i><font color="Green" title="E45">A18</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E46">A19</font></i>: 
<font color="Maroon" title="c5">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E5:54"><i><font color="Green" title="E41">A14</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_2"/><i><font color="Green" title="E47">A20</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>set </b><font color="Maroon" title="c9">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> ;<br/><a NAME="E3:54_2"/><i><font color="Green" title="E48">A21</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_2"><i><font color="Green" title="E46">A19</font></i></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" title="c10">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2_1"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c11">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green" title="E49">A22</font></i>: 
<font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">b</font>)
 <b>and </b><br/><a NAME="E4:54_2_1"/><i><font color="Green" title="E50">A23</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_2_1"/>
<font color="Maroon" title="c10">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_2"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="dickson.html#E3:54_2_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="dickson.html#E4:54_2_1"><i><font color="Green" title="E50">A23</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_2"/><b>then </b><i><font color="Green" title="E49">A24</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:54_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:54_2_2"/><b>then </b>
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">s</font>
 ;<br/><a NAME="E5:54_2_2"/><b>then </b><i><font color="Green" title="E50">A25</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">sx</font> = <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>consider </b><font color="Maroon" title="c13">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E8:54_2_2"/><i><font color="Green" title="E51">A26</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E9:54_2_2"/><i><font color="Green" title="E52">A27</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">sx</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="dickson.html#E5:54_2_2"><i><font color="Green" title="E50">A25</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/><a NAME="E10:54_2_2"/>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c13">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> <b>by </b><i><a class="txt" href="dickson.html#E8:54_2_2"><i><font color="Green" title="E51">A26</font></i></a></i>;<br/><a NAME="E12:54_2_2"/><i><font color="Green" title="E53">A28</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">b</font>)
 <b>by </b><i><a class="txt" href="dickson.html#E9:54_2_2"><i><font color="Green" title="E52">A27</font></i></a></i>;<br/><a NAME="E13:54_2_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">b</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E8:54_2_2"><i><font color="Green" title="E51">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E14:54_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E12:54_2_2"><i><font color="Green" title="E53">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E7:54_2"/><b>then </b>
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E8:54_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E5:54_2"><i><font color="Green" title="E49">A24</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c9">tb</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E14:54"/><i><font color="Green" title="E46">A29</font></i>: 
( <font color="Maroon" title="c9">tb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">tb</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<a NAME="E15:54"/><i><font color="Green" title="E47">A30</font></i>: 
<font color="Maroon" title="c9">tb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E14:54"><i><font color="Green" title="E46">A29</font></i></a></i>;<br/>
<a NAME="E16:54"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">bt</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font><b> such that </b><br/><a NAME="E18:54"/><i><font color="Green" title="E48">A31</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">bt</font>
 <b>and </b><br/><a NAME="E19:54"/><i><font color="Green" title="E49">A32</font></i>: 
 ex <font color="Olive" title="b1">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">bt</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">b'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E21:54"/><i><font color="Green" title="E50">A33</font></i>: 
<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">bt</font>
 <b>and </b><br/><a NAME="E22:54"/>
not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E19:54"><i><font color="Green" title="E49">A32</font></i></a></i>;<br/>
<a NAME="E23:54"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17" title="NORMSP_1:th.17">NORMSP_1:17</a></i>;<br/>
<a NAME="E24:54"/><b>then </b><i><font color="Green" title="E51">A34</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E8:54"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="dickson.html#E18:54"><i><font color="Green" title="E48">A31</font></i></a>, <a class="txt" href="dickson.html#E21:54"><i><font color="Green" title="E50">A33</font></i></a>, <a class="ref" href="dickson.html#D11" target="_self" title="DICKSON:def.11">Def11</a></i>;<br/>
<a NAME="E25:54"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E15:54"><i><font color="Green" title="E47">A30</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">eb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font><b> such that </b><br/><a NAME="E27:54"/><i><font color="Green" title="E52">A35</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">eb</font>
 <b>and </b><br/><a NAME="E28:54"/><i><font color="Green" title="E53">A36</font></i>: 
 ex <font color="Olive" title="b1">eb'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b1">eb'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">eb</font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">eb'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">eb'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E30:54"/><i><font color="Green" title="E54">A37</font></i>: 
<font color="Maroon" title="c13">eb'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">eb</font>
 <b>and </b><br/><a NAME="E31:54"/><i><font color="Green" title="E55">A38</font></i>: 
not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c13">eb'</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E28:54"><i><font color="Green" title="E53">A36</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<b>set </b><font color="Maroon" title="c14">W1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c15">W2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c16">W3</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span> ;<br/>
<a NAME="E32:54"/><i><font color="Green" title="E56">A39</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="finseq_1.html#S6" title="FINSEQ_1:sch.6">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" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_3_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c18">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_3_1"/><i><font color="Green" title="E57">A40</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w</font>
 <b>and </b><br/><a NAME="E4:54_3_1"/><i><font color="Green" title="E58">A41</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">w</font>
 ;<br/><a NAME="E5:54_3_1"/><i><font color="Green" title="E59">A42</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E6:54_3_1"/>
( <font color="Maroon" title="c18">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c18">w</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E7:54_3_1"/><b>then </b>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_3_1"><i><font color="Green" title="E57">A40</font></i></a>, <a class="txt" href="dickson.html#E4:54_3_1"><i><font color="Green" title="E58">A41</font></i></a>, <a class="txt" href="dickson.html#E5:54_3_1"><i><font color="Green" title="E59">A42</font></i></a></i>;<br/><b>hence </b><a NAME="E8:54_3_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_3"/><i><font color="Green" title="E57">A43</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">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" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="dickson.html#E2:54_3"><i><font color="Green" title="E57">A43</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_3_2_1"/><i><font color="Green" title="E58">A44</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w</font> &amp; <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_3_2_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_3_2_1"><i><font color="Green" title="E58">A44</font></i></a></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" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_3_2_2"/><i><font color="Green" title="E58">A45</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">w</font> &amp; <font color="Maroon" title="c18">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_3_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_3_2_2"><i><font color="Green" title="E58">A45</font></i></a></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" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E35:54"/><b>then </b><i><font color="Green" title="E57">A46</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E24:54"><i><font color="Green" title="E51">A34</font></i></a>, <a class="txt" href="dickson.html#E27:54"><i><font color="Green" title="E52">A35</font></i></a>, <a class="txt" href="dickson.html#E30:54"><i><font color="Green" title="E54">A37</font></i></a>, <a class="txt" href="dickson.html#E31:54"><i><font color="Green" title="E55">A38</font></i></a>, <a class="txt" href="dickson.html#E32:54"><i><font color="Green" title="E56">A39</font></i></a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon">S<sub>1</sub></font>[$1,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>];<br/>
<a NAME="E36:54"/><i><font color="Green" title="E58">A47</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="dickson.html#E35:54"><i><font color="Green" title="E57">A46</font></i></a></i>;<br/>
<div><i><font color="Green" title="E59">A48</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" title="c17">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:54_4"/><i><font color="Green" title="E60">A49</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">k</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c18">N</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c19">B</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:54_4"/><i><font color="Green" title="E61">A50</font></i>: 
<font color="Maroon" title="c18">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:54_4"/><i><font color="Green" title="E62">A51</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E5:54_4"/><i><font color="Green" title="E63">A52</font></i>: 
<font color="Maroon" title="c19">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E6:54_4"/><i><font color="Green" title="E64">A53</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54_4"><i><font color="Green" title="E60">A49</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c20">BALL</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c21">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c22">iN</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c23">BD</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:54_4"/><i><font color="Green" title="E65">A54</font></i>: 
( <font color="Maroon" title="c23">BD</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c23">BD</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/><a NAME="E9:54_4"/>
<font color="Maroon" title="c23">BD</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E8:54_4"><i><font color="Green" title="E65">A54</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></i>;<br/><a NAME="E10:54_4"/><b>then </b>
<font color="Maroon" title="c23">BD</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E8:54_4"><i><font color="Green" title="E65">A54</font></i></a></i>;<br/><a NAME="E11:54_4"/><b>then </b>
<font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E5:54_4"><i><font color="Green" title="E63">A52</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">BB</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E13:54_4"/><i><font color="Green" title="E66">A55</font></i>: 
( <font color="Maroon" title="c19">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">BB</font> &amp; <font color="Maroon" title="c24">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><a NAME="E14:54_4"/><i><font color="Green" title="E67">A56</font></i>: 
<font color="Maroon" title="c19">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c18">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E13:54_4"><i><font color="Green" title="E66">A55</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</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" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">H<sub>1</sub></font>($1,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>);<br/><a NAME="E1:54_4_1"/><i><font color="Green" title="E68">A57</font></i>: 
<font color="Maroon" title="c19">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E13:54_4"><i><font color="Green" title="E66">A55</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_4_1"/><i><font color="Green" title="E69">A58</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>set </b><font color="Maroon" title="c25">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> </span>}</span> ;<br/><a NAME="E3:54_4_1"/><i><font color="Green" title="E70">A59</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_4_1"><i><font color="Green" title="E68">A57</font></i></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" title="c26">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_4_1_1"/>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c27">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E3:54_4_1_1"/><i><font color="Green" title="E71">A60</font></i>: 
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c27">b</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>)
 <b>and </b><br/><a NAME="E4:54_4_1_1"/><i><font color="Green" title="E72">A61</font></i>: 
<font color="Maroon" title="c27">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_4_1_1"/>
<font color="Maroon" title="c26">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_4_1"><i><font color="Green" title="E69">A58</font></i></a>, <a class="txt" href="dickson.html#E3:54_4_1_1"><i><font color="Green" title="E71">A60</font></i></a>, <a class="txt" href="dickson.html#E4:54_4_1_1"><i><font color="Green" title="E72">A61</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_4_1"/><b>then </b><i><font color="Green" title="E71">A62</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_1"><i><font color="Green" title="E70">A59</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_4_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">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" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_4_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c27">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_1_2"/><i><font color="Green" title="E72">A63</font></i>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">w</font>
 <b>and </b><br/><a NAME="E4:54_4_1_2"/><i><font color="Green" title="E73">A64</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">w</font>
 <b>and </b><br/><a NAME="E5:54_4_1_2"/><i><font color="Green" title="E74">A65</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">w</font>
 ;<br/>
<a NAME="E6:54_4_1_2"/><i><font color="Green" title="E75">A66</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:54_4"><i><font color="Green" title="E61">A50</font></i></a>, <a class="txt" href="dickson.html#E4:54_4_1_2"><i><font color="Green" title="E73">A64</font></i></a>, <a class="txt" href="dickson.html#E5:54_4_1_2"><i><font color="Green" title="E74">A65</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">sw</font> = <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>consider </b><font color="Maroon" title="c29">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E9:54_4_1_2"/><i><font color="Green" title="E76">A67</font></i>: 
<font color="Maroon" title="c29">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font>
 <b>and </b><br/><a NAME="E10:54_4_1_2"/><i><font color="Green" title="E77">A68</font></i>: 
<font color="Maroon" title="c29">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">sw</font>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_4"><i><font color="Green" title="E67">A56</font></i></a>, <a class="txt" href="dickson.html#E6:54_4_1_2"><i><font color="Green" title="E75">A66</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E11:54_4_1_2"/><i><font color="Green" title="E78">A69</font></i>: 
<font color="Maroon" title="c19">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E14:54_4"><i><font color="Green" title="E67">A56</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E12:54_4_1_2"/>
<font color="Maroon" title="c18">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">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" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_4_1_2_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_1_2_1"/><i><font color="Green" title="E79">A70</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">u</font>
 <b>and </b><br/><a NAME="E4:54_4_1_2_1"/>
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">u</font>
 <b>and </b><br/><a NAME="E5:54_4_1_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">u</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4"><i><font color="Green" title="E61">A50</font></i></a></i>;<br/>
<a NAME="E6:54_4_1_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_4_1_2_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_1_2_1"><i><font color="Green" title="E79">A70</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_4_1_2"/><b>then </b>
<font color="Maroon" title="c19">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E11:54_4_1_2"><i><font color="Green" title="E78">A69</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">b</font> = <font color="Maroon" title="c29">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> <b>by </b><i><a class="txt" href="dickson.html#E9:54_4_1_2"><i><font color="Green" title="E76">A67</font></i></a></i>;<br/>
<a NAME="E15:54_4_1_2"/><i><font color="Green" title="E79">A71</font></i>: 
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c30">b</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>)
 
<b>by </b><i><a class="txt" href="dickson.html#E5:54_4_1_2"><i><font color="Green" title="E74">A65</font></i></a>, <a class="txt" href="dickson.html#E10:54_4_1_2"><i><font color="Green" title="E77">A68</font></i></a></i>;<br/>
<a NAME="E16:54_4_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c30">b</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54_4_1_2"><i><font color="Green" title="E76">A67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:54_4_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_1_2"><i><font color="Green" title="E72">A63</font></i></a>, <a class="txt" href="dickson.html#E15:54_4_1_2"><i><font color="Green" title="E79">A71</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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="txt" href="dickson.html#E4:54_4"><i><font color="Green" title="E62">A51</font></i></a>, <a class="txt" href="dickson.html#E5:54_4_1"><i><font color="Green" title="E71">A62</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c25">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E17:54_4"/><i><font color="Green" title="E68">A72</font></i>: 
( <font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">b</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><a NAME="E18:54_4"/>
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E17:54_4"><i><font color="Green" title="E68">A72</font></i></a></i>;<br/><a NAME="E19:54_4"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c26">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font><b> such that </b><br/><a NAME="E21:54_4"/><i><font color="Green" title="E69">A73</font></i>: 
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 <b>and </b><br/><a NAME="E22:54_4"/><i><font color="Green" title="E70">A74</font></i>: 
 ex <font color="Olive" title="b1">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><b>consider </b><font color="Maroon" title="c27">b'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E24:54_4"/><i><font color="Green" title="E71">A75</font></i>: 
<font color="Maroon" title="c27">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">b</font>
 <b>and </b><br/><a NAME="E25:54_4"/><i><font color="Green" title="E72">A76</font></i>: 
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c27">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E22:54_4"><i><font color="Green" title="E70">A74</font></i></a></i>;<br/><a NAME="E26:54_4"/><i><font color="Green" title="E73">A77</font></i>: 
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font>
 ;<br/><a NAME="E27:54_4"/>
<font color="Maroon" title="c19">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_4"><i><font color="Green" title="E67">A56</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/><a NAME="E28:54_4"/><b>then </b>
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E26:54_4"><i><font color="Green" title="E73">A77</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E30:54_4"/><i><font color="Green" title="E74">A78</font></i>: 
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">w</font>
 <b>and </b><br/><a NAME="E31:54_4"/>
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">w</font>
 <b>and </b><br/><a NAME="E32:54_4"/><i><font color="Green" title="E75">A79</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">w</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4"><i><font color="Green" title="E61">A50</font></i></a></i>;<br/><a NAME="E33:54_4"/><i><font color="Green" title="E76">A80</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c19">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E6:54_4"><i><font color="Green" title="E64">A53</font></i></a>, <a class="txt" href="dickson.html#E21:54_4"><i><font color="Green" title="E69">A73</font></i></a>, <a class="txt" href="dickson.html#E30:54_4"><i><font color="Green" title="E74">A78</font></i></a>, <a class="txt" href="dickson.html#E32:54_4"><i><font color="Green" title="E75">A79</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></i>;<br/><a NAME="E34:54_4"/><i><font color="Green" title="E77">A81</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E6:54_4"><i><font color="Green" title="E64">A53</font></i></a>, <a class="txt" href="dickson.html#E21:54_4"><i><font color="Green" title="E69">A73</font></i></a>, <a class="txt" href="dickson.html#E30:54_4"><i><font color="Green" title="E74">A78</font></i></a>, <a class="txt" href="dickson.html#E32:54_4"><i><font color="Green" title="E75">A79</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/><b>set </b><font color="Maroon" title="c29">W1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c30">W2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c31">W3</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">w1</font>] ) </span>}</span> ;<br/><a NAME="E35:54_4"/><i><font color="Green" title="E78">A82</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">w1</font>] ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="dickson.html#S1" target="_self" title="DICKSON:sch.1">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" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_4_2_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c33">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_2_1"/><i><font color="Green" title="E79">A83</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">w</font>
 <b>and </b><br/><a NAME="E4:54_4_2_1"/><i><font color="Green" title="E80">A84</font></i>: 
( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">w</font> )
 ;<br/><a NAME="E5:54_4_2_1"/>
( <font color="Maroon" title="c33">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c33">w</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ;<br/><a NAME="E6:54_4_2_1"/><b>then </b>
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span>  or <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">w1</font>] ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_2_1"><i><font color="Green" title="E79">A83</font></i></a>, <a class="txt" href="dickson.html#E4:54_4_2_1"><i><font color="Green" title="E80">A84</font></i></a></i>;<br/><b>hence </b><a NAME="E7:54_4_2_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">w1</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="E79">A85</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">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" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span>  or <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">w1</font>] ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="dickson.html#E2:54_4_2"><i><font color="Green" title="E79">A85</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c33">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_2_2_1"/><i><font color="Green" title="E80">A86</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">w</font> &amp; <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">w</font> )
 ;<br/><a NAME="E4:54_4_2_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">w</font>
 <b>by </b><i><a class="txt" href="dickson.html#E34:54_4"><i><font color="Green" title="E77">A81</font></i></a>, <a class="txt" href="dickson.html#E3:54_4_2_2_1"><i><font color="Green" title="E80">A86</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:54_4_2_2_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_2_2_1"><i><font color="Green" title="E80">A86</font></i></a></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" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">w1</font>] ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c33">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_4_2_2_2"/><i><font color="Green" title="E80">A87</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">w</font> &amp; <font color="Maroon" title="c33">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">w</font> )
 ;<br/><b>thus </b><a NAME="E4:54_4_2_2_2"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_2_2_2"><i><font color="Green" title="E80">A87</font></i></a></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" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">w1</font>) where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">w1</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E38:54_4"/><b>then </b>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">w1</font> where <font color="Olive" title="b1">w1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E21:54_4"><i><font color="Green" title="E69">A73</font></i></a>, <a class="txt" href="dickson.html#E24:54_4"><i><font color="Green" title="E71">A75</font></i></a>, <a class="txt" href="dickson.html#E25:54_4"><i><font color="Green" title="E72">A76</font></i></a>, <a class="txt" href="dickson.html#E33:54_4"><i><font color="Green" title="E76">A80</font></i></a>, <a class="txt" href="dickson.html#E35:54_4"><i><font color="Green" title="E78">A82</font></i></a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">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" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="dickson.html#E9:54"><i><font color="Green" title="E43">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E38:54"/><i><font color="Green" title="E60">A88</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="dickson.html#E36:54"><i><font color="Green" title="E58">A47</font></i></a>, <a class="txt" href="dickson.html#E37:54"><i><font color="Green" title="E59">A48</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c17">t</font> = <font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font>;<br/>
<b>take </b>
<font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">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" title="c19">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:54_5_1"/>
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f</font>
 <b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">fn</font> = <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="dickson.html#E11:54"><i><font color="Green" title="E45">A18</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">N</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c22">B</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:54_5_1"/><i><font color="Green" title="E61">A89</font></i>: 
<font color="Maroon" title="c21">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c20">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E5:54_5_1"/><i><font color="Green" title="E62">A90</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c20">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E6:54_5_1"/><i><font color="Green" title="E63">A91</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E7:54_5_1"/><i><font color="Green" title="E64">A92</font></i>: 
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon" title="c20">fn</font>
 <b>by </b><i><a class="txt" href="dickson.html#E38:54"><i><font color="Green" title="E60">A88</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c23">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c24">BJ</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c25">BC</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c26">BD</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:54_5_1"/><i><font color="Green" title="E65">A93</font></i>: 
( <font color="Maroon" title="c26">BD</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c26">BD</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/><a NAME="E10:54_5_1"/>
<font color="Maroon" title="c26">BD</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E9:54_5_1"><i><font color="Green" title="E65">A93</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></i>;<br/><a NAME="E11:54_5_1"/><b>then </b>
<font color="Maroon" title="c26">BD</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E9:54_5_1"><i><font color="Green" title="E65">A93</font></i></a></i>;<br/><a NAME="E12:54_5_1"/><b>then </b>
<font color="Maroon" title="c22">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E6:54_5_1"><i><font color="Green" title="E63">A91</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">BB</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E14:54_5_1"/><i><font color="Green" title="E66">A94</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">BB</font>
 <b>and </b><br/><a NAME="E15:54_5_1"/><i><font color="Green" title="E67">A95</font></i>: 
<font color="Maroon" title="c27">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><a NAME="E16:54_5_1"/><i><font color="Green" title="E68">A96</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E14:54_5_1"><i><font color="Green" title="E66">A94</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></i>;<br/><a NAME="E17:54_5_1"/><b>then </b><i><font color="Green" title="E69">A97</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">N</font>
 <b>by </b><i><a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</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" title="E70">A98</font></i>: 
<font color="Maroon" title="c22">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_5_1"><i><font color="Green" title="E66">A94</font></i></a>, <a class="txt" href="dickson.html#E15:54_5_1"><i><font color="Green" title="E67">A95</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_5_1_1"/><i><font color="Green" title="E71">A99</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">H<sub>1</sub></font>($1,<font color="Maroon" title="c20">fn</font>);<br/><b>set </b><font color="Maroon" title="c28">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c29">iN</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c20">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> ;<br/><a NAME="E3:54_5_1_1"/><i><font color="Green" title="E72">A100</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_5_1_1"><i><font color="Green" title="E70">A98</font></i></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" title="c30">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_5_1_1_1"/>
<font color="Maroon" title="c30">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c31">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E3:54_5_1_1_1"/><i><font color="Green" title="E73">A101</font></i>: 
<font color="Maroon" title="c30">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c31">b</font>,<font color="Maroon" title="c20">fn</font>)
 <b>and </b><br/><a NAME="E4:54_5_1_1_1"/><i><font color="Green" title="E74">A102</font></i>: 
<font color="Maroon" title="c31">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_5_1_1_1"/>
<font color="Maroon" title="c30">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_5_1_1"><i><font color="Green" title="E71">A99</font></i></a>, <a class="txt" href="dickson.html#E3:54_5_1_1_1"><i><font color="Green" title="E73">A101</font></i></a>, <a class="txt" href="dickson.html#E4:54_5_1_1_1"><i><font color="Green" title="E74">A102</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_5_1_1"/><b>then </b><i><font color="Green" title="E73">A103</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_5_1_1"><i><font color="Green" title="E72">A100</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_5_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c20">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">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" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_5_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c20">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_5_1_1_2"/><i><font color="Green" title="E74">A104</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">w</font>
 <b>and </b><br/><a NAME="E4:54_5_1_1_2"/><i><font color="Green" title="E75">A105</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">w</font>
 <b>and </b><br/><a NAME="E5:54_5_1_1_2"/><i><font color="Green" title="E76">A106</font></i>: 
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">w</font>
 ;<br/>
<a NAME="E6:54_5_1_1_2"/><i><font color="Green" title="E77">A107</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E4:54_5_1"><i><font color="Green" title="E61">A89</font></i></a>, <a class="txt" href="dickson.html#E4:54_5_1_1_2"><i><font color="Green" title="E75">A105</font></i></a>, <a class="txt" href="dickson.html#E5:54_5_1_1_2"><i><font color="Green" title="E76">A106</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">sw</font> = <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>consider </b><font color="Maroon" title="c33">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E9:54_5_1_1_2"/><i><font color="Green" title="E78">A108</font></i>: 
<font color="Maroon" title="c33">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>and </b><br/><a NAME="E10:54_5_1_1_2"/><i><font color="Green" title="E79">A109</font></i>: 
<font color="Maroon" title="c33">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c32">sw</font>
 <b>by </b><i><a class="txt" href="dickson.html#E16:54_5_1"><i><font color="Green" title="E68">A96</font></i></a>, <a class="txt" href="dickson.html#E6:54_5_1_1_2"><i><font color="Green" title="E77">A107</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E11:54_5_1_1_2"/><i><font color="Green" title="E80">A110</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E16:54_5_1"><i><font color="Green" title="E68">A96</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E12:54_5_1_1_2"/>
<font color="Maroon" title="c21">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">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" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_5_1_1_2_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c35">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_5_1_1_2_1"/><i><font color="Green" title="E81">A111</font></i>: 
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">u</font>
 <b>and </b><br/><a NAME="E4:54_5_1_1_2_1"/>
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">u</font>
 <b>and </b><br/><a NAME="E5:54_5_1_1_2_1"/>
<font color="Maroon" title="c20">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">u</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_5_1"><i><font color="Green" title="E61">A89</font></i></a></i>;<br/>
<a NAME="E6:54_5_1_1_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_5_1_1_2_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_5_1_1_2_1"><i><font color="Green" title="E81">A111</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">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" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E11:54_5_1_1_2"><i><font color="Green" title="E80">A110</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c34">b</font> = <font color="Maroon" title="c33">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> <b>by </b><i><a class="txt" href="dickson.html#E9:54_5_1_1_2"><i><font color="Green" title="E78">A108</font></i></a></i>;<br/>
<a NAME="E15:54_5_1_1_2"/><i><font color="Green" title="E81">A112</font></i>: 
<font color="Maroon" title="c31">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">b</font>,<font color="Maroon" title="c20">fn</font>)
 
<b>by </b><i><a class="txt" href="dickson.html#E5:54_5_1_1_2"><i><font color="Green" title="E76">A106</font></i></a>, <a class="txt" href="dickson.html#E10:54_5_1_1_2"><i><font color="Green" title="E79">A109</font></i></a></i>;<br/>
<a NAME="E16:54_5_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">b</font>,<font color="Maroon" title="c20">fn</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54_5_1_1_2"><i><font color="Green" title="E78">A108</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:54_5_1_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_5_1_1_2"><i><font color="Green" title="E74">A104</font></i></a>, <a class="txt" href="dickson.html#E15:54_5_1_1_2"><i><font color="Green" title="E81">A112</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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="txt" href="dickson.html#E5:54_5_1"><i><font color="Green" title="E62">A90</font></i></a>, <a class="txt" href="dickson.html#E5:54_5_1_1"><i><font color="Green" title="E73">A103</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c28">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E20:54_5_1"/><i><font color="Green" title="E70">A113</font></i>: 
( <font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c28">b</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><a NAME="E21:54_5_1"/>
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E20:54_5_1"><i><font color="Green" title="E70">A113</font></i></a></i>;<br/><a NAME="E22:54_5_1"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c29">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font><b> such that </b><br/><a NAME="E24:54_5_1"/><i><font color="Green" title="E71">A114</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">b</font>
 <b>and </b><br/><a NAME="E25:54_5_1"/>
 ex <font color="Olive" title="b1">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/><a NAME="E26:54_5_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E17:54_5_1"><i><font color="Green" title="E69">A97</font></i></a>, <a class="txt" href="dickson.html#E24:54_5_1"><i><font color="Green" title="E71">A114</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E28:54_5_1"/><i><font color="Green" title="E72">A115</font></i>: 
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c20">fn</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">j</font> &amp; <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">j</font> &amp; <font color="Maroon" title="c20">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">j</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_5_1"><i><font color="Green" title="E61">A89</font></i></a></i>;<br/><b>thus </b><a NAME="E29:54_5_1"/>
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E7:54_5_1"><i><font color="Green" title="E64">A92</font></i></a>, <a class="txt" href="dickson.html#E28:54_5_1"><i><font color="Green" title="E72">A115</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:54_5"/>
<font color="Maroon" title="c18">f</font> is <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D11" title="SEQM_3:def.11">SEQM_3:def 11</a></i>;<br/><b>let </b><font color="Maroon" title="c19">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E3:54_5"/>
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f</font>
 <b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:54_5"/>
<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="dickson.html#E11:54"><i><font color="Green" title="E45">A18</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c19">f</font> = <font color="Maroon" title="c18">f</font> as  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a> ;<br/>
<a NAME="E42:54"/>
<font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">f</font>
 
;<br/>
<b>hence </b><a NAME="E43:54"/>
<font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D4" title="BHSP_3:def.4">BHSP_3:def 4</a></i>;<br/>

<b>let </b><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D2" target="_self" title="DICKSON:def.2">DICKSON:def 2</a><br/>

<a NAME="E44:54"/><i><font color="Green" title="E61">A116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E45:54"/><i><font color="Green" title="E62">A117</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">N</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c22">B</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E47:54"/><i><font color="Green" title="E63">A118</font></i>: 
<font color="Maroon" title="c21">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E48:54"/><i><font color="Green" title="E64">A119</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E49:54"/><i><font color="Green" title="E65">A120</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E50:54"/><i><font color="Green" title="E66">A121</font></i>: 
<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="dickson.html#K8" title="DICKSON:func.8">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E38:54"><i><font color="Green" title="E60">A88</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">BX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c24">sfn1</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c25">BBX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c26">BD</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E52:54"/><i><font color="Green" title="E67">A122</font></i>: 
( <font color="Maroon" title="c26">BD</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c26">BD</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/>
<a NAME="E53:54"/>
<font color="Maroon" title="c26">BD</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E52:54"><i><font color="Green" title="E67">A122</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></i>;<br/>
<a NAME="E54:54"/><b>then </b>
<font color="Maroon" title="c26">BD</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E52:54"><i><font color="Green" title="E67">A122</font></i></a></i>;<br/>
<a NAME="E55:54"/><b>then </b>
<font color="Maroon" title="c22">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">BB</font> where <font color="Olive" title="b1">BB</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E49:54"><i><font color="Green" title="E65">A120</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c27">BB</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dickson.html#K9" title="DICKSON:func.9">Dickson-bases</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E57:54"/><i><font color="Green" title="E68">A123</font></i>: 
<font color="Maroon" title="c27">BB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">B</font>
 <b>and </b><br/><a NAME="E58:54"/><i><font color="Green" title="E69">A124</font></i>: 
<font color="Maroon" title="c27">BB</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>
<a NAME="E59:54"/><i><font color="Green" title="E70">A125</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c21">N</font>,<font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="dickson.html#E57:54"><i><font color="Green" title="E68">A123</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</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" title="E71">A126</font></i>: 
<font color="Maroon" title="c22">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E57:54"><i><font color="Green" title="E68">A123</font></i></a>, <a class="txt" href="dickson.html#E58:54"><i><font color="Green" title="E69">A124</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_6"/><i><font color="Green" title="E72">A127</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">H<sub>1</sub></font>($1,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>);<br/><b>set </b><font color="Maroon" title="c28">Ball</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c29">iN</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> ;<br/><a NAME="E3:54_6"/><i><font color="Green" title="E73">A128</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_6"><i><font color="Green" title="E71">A126</font></i></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" title="c30">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_6_1"/>
<font color="Maroon" title="c30">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c31">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E3:54_6_1"/><i><font color="Green" title="E74">A129</font></i>: 
<font color="Maroon" title="c30">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c31">b</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>)
 <b>and </b><br/><a NAME="E4:54_6_1"/><i><font color="Green" title="E75">A130</font></i>: 
<font color="Maroon" title="c31">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 ;<br/><b>thus </b><a NAME="E5:54_6_1"/>
<font color="Maroon" title="c30">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_6"><i><font color="Green" title="E72">A127</font></i></a>, <a class="txt" href="dickson.html#E3:54_6_1"><i><font color="Green" title="E74">A129</font></i></a>, <a class="txt" href="dickson.html#E4:54_6_1"><i><font color="Green" title="E75">A130</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_6"/><b>then </b><i><font color="Green" title="E74">A131</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_6"><i><font color="Green" title="E73">A128</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_6"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">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" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_6_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c31">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_6_2"/><i><font color="Green" title="E75">A132</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">w</font>
 <b>and </b><br/><a NAME="E4:54_6_2"/><i><font color="Green" title="E76">A133</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">w</font>
 <b>and </b><br/><a NAME="E5:54_6_2"/><i><font color="Green" title="E77">A134</font></i>: 
<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">w</font>
 ;<br/>
<a NAME="E6:54_6_2"/><i><font color="Green" title="E78">A135</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E47:54"><i><font color="Green" title="E63">A118</font></i></a>, <a class="txt" href="dickson.html#E4:54_6_2"><i><font color="Green" title="E76">A133</font></i></a>, <a class="txt" href="dickson.html#E5:54_6_2"><i><font color="Green" title="E77">A134</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">sw</font> = <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>consider </b><font color="Maroon" title="c33">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E9:54_6_2"/><i><font color="Green" title="E79">A136</font></i>: 
<font color="Maroon" title="c33">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>and </b><br/><a NAME="E10:54_6_2"/><i><font color="Green" title="E80">A137</font></i>: 
<font color="Maroon" title="c33">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c32">sw</font>
 <b>by </b><i><a class="txt" href="dickson.html#E59:54"><i><font color="Green" title="E70">A125</font></i></a>, <a class="txt" href="dickson.html#E6:54_6_2"><i><font color="Green" title="E78">A135</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E11:54_6_2"/><i><font color="Green" title="E81">A138</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E59:54"><i><font color="Green" title="E70">A125</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E12:54_6_2"/>
<font color="Maroon" title="c21">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">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" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_6_2_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c35">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_6_2_1"/><i><font color="Green" title="E82">A139</font></i>: 
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">u</font>
 <b>and </b><br/><a NAME="E4:54_6_2_1"/>
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">u</font>
 <b>and </b><br/><a NAME="E5:54_6_2_1"/>
<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">u</font>
 <b>by </b><i><a class="txt" href="dickson.html#E47:54"><i><font color="Green" title="E63">A118</font></i></a></i>;<br/>
<a NAME="E6:54_6_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_6_2_1"/>
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_6_2_1"><i><font color="Green" title="E82">A139</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_6_2"/><b>then </b>
<font color="Maroon" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E11:54_6_2"><i><font color="Green" title="E81">A138</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c34">b</font> = <font color="Maroon" title="c33">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> <b>by </b><i><a class="txt" href="dickson.html#E9:54_6_2"><i><font color="Green" title="E79">A136</font></i></a></i>;<br/>
<a NAME="E15:54_6_2"/><i><font color="Green" title="E82">A140</font></i>: 
<font color="Maroon" title="c31">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">b</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>)
 
<b>by </b><i><a class="txt" href="dickson.html#E5:54_6_2"><i><font color="Green" title="E77">A134</font></i></a>, <a class="txt" href="dickson.html#E10:54_6_2"><i><font color="Green" title="E80">A137</font></i></a></i>;<br/>
<a NAME="E16:54_6_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c34">b</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54_6_2"><i><font color="Green" title="E79">A136</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:54_6_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_6_2"><i><font color="Green" title="E75">A132</font></i></a>, <a class="txt" href="dickson.html#E15:54_6_2"><i><font color="Green" title="E82">A140</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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#E48:54"><i><font color="Green" title="E64">A119</font></i></a>, <a class="txt" href="dickson.html#E5:54_6"><i><font color="Green" title="E74">A131</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c28">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font><b> such that </b><br/><a NAME="E62:54"/><i><font color="Green" title="E71">A141</font></i>: 
( <font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c28">b</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<a NAME="E63:54"/>
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E62:54"><i><font color="Green" title="E71">A141</font></i></a></i>;<br/>
<a NAME="E64:54"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c29">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font><b> such that </b><br/><a NAME="E66:54"/><i><font color="Green" title="E72">A142</font></i>: 
<font color="Maroon" title="c29">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span> 
 <b>and </b><br/><a NAME="E67:54"/>
 ex <font color="Olive" title="b1">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<a NAME="E68:54"/><i><font color="Green" title="E73">A143</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E66:54"><i><font color="Green" title="E72">A142</font></i></a></i>;<br/>
<a NAME="E69:54"/>
<font color="Maroon" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E59:54"><i><font color="Green" title="E70">A125</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E70:54"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E68:54"><i><font color="Green" title="E73">A143</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c30">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E72:54"/><i><font color="Green" title="E74">A144</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">B</font> :  ex <font color="Olive" title="b2">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b'</font>,<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font>) is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">w</font>
 <b>and </b><br/><a NAME="E73:54"/><i><font color="Green" title="E75">A145</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">w</font>
 <b>and </b><br/><a NAME="E74:54"/><i><font color="Green" title="E76">A146</font></i>: 
<font color="Maroon" title="c19">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">w</font>
 <b>by </b><i><a class="txt" href="dickson.html#E47:54"><i><font color="Green" title="E63">A118</font></i></a></i>;<br/>
<a NAME="E75:54"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E44:54"><i><font color="Green" title="E61">A116</font></i></a>, <a class="txt" href="dickson.html#E45:54"><i><font color="Green" title="E62">A117</font></i></a>, <a class="txt" href="dickson.html#E50:54"><i><font color="Green" title="E66">A121</font></i></a>, <a class="txt" href="dickson.html#E72:54"><i><font color="Green" title="E74">A144</font></i></a>, <a class="txt" href="dickson.html#E73:54"><i><font color="Green" title="E75">A145</font></i></a>, <a class="txt" href="dickson.html#E74:54"><i><font color="Green" title="E76">A146</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E76:54"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


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