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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font>  st <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  holds <br/> ex <font color="Olive" title="b2">t</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">t</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</font> is  <a href="dickson.html#V2" title="DICKSON:attr.2">weakly-ascending</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  implies  ex <font color="Olive" title="b1">t</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">t</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c2">s</font> &amp; <font color="Olive" title="b1">t</font> is  <a href="dickson.html#V2" title="DICKSON:attr.2">weakly-ascending</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">t</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c2">s</font> &amp; <font color="Olive" title="b1">t</font> is  <a href="dickson.html#V2" title="DICKSON:attr.2">weakly-ascending</a>  )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">deffunc </span><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#K2" title="RELSET_1:func.2">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> ) <span class="kw">-&gt; </span>   <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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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/>
<span class="kw">deffunc </span><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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>) <span class="kw">-&gt; </span>   <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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">n</font> </span> } </span> ;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> $2 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> $2 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b4">b9</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">b9</font>,$2) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> </span>)</span>,$2) );<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>(  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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> $2 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<span class="kw">set </span><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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>  implies <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> )</span><br/><span class="kw">assume </span><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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><a NAME="E2:54_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">set </span><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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:54_1_2_1"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c7">N</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,1]</span><br/><span class="kw">assume </span><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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,1]</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">ww</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:54_1_2_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">consider </span><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> <span class="kw"> such that </span><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>
 <span class="kw">and </span><br/><a NAME="E7:54_1_2_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c10">www</font>
 <span class="kw">and </span><br/><a NAME="E8:54_1_2_1"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_2_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a></span>;<br/><a NAME="E9:54_1_2_1"/>
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1_2_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E8:54_1_2_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:54_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]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_2_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:54_1_2_2"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
 not <font color="Maroon" title="c7">N</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">set </span><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/><span class="kw">set </span><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/><span class="kw">consider </span><font color="Maroon" title="c11">BD1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:54_1_2_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:54_1_2_2"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c11">BD1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></span>;<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E3:54_1_2_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/><a NAME="E6:54_1_2_2"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_2_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>;<br/><a NAME="E7:54_1_2_2"/><span class="kw">then </span>
 <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/><a NAME="E8:54_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>) st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">BB</font> where <font color="Olive" title="b2">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="b2">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="Olive" title="b1">BBB</font> &amp; <font color="Olive" title="b1">BBB</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 ;<br/><a NAME="E9:54_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/><a NAME="E10:54_1_2_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_2_2"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E9:54_1_2_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="dickson.html#T26" target="_self" title="DICKSON:th.26">Th27</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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/><span class="kw">take </span>
<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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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>)]</span><br/><span class="kw">set </span><font color="Maroon" title="c14">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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/><span class="kw">assume </span><a NAME="E12:54_1_2_2"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
 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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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>)]</span><br/><span class="kw">take </span>
<font color="Maroon" title="c7">N</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b2">w</font></span>)</span> where <font color="Olive" title="b2">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b2">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="b2">w</font> ) </span> } </span>  &amp;  not <span class="p1"> { <span class="default"> <font color="Olive" title="b2">w</font> where <font color="Olive" title="b2">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b2">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="b2">w</font> ) </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Olive" title="b1">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="b2">BB</font> where <font color="Olive" title="b2">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="b2">BB</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  </span> } </span>  &amp; <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="b2">b</font> where <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">B</font> :  ex <font color="Olive" title="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">b9</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="b2">b</font> where <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">B</font> :  ex <font color="Olive" title="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">b9</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>) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">B</font> = <font color="Maroon" title="c12">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/><span class="kw">take </span>
<font color="Maroon" title="c15">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>  &amp;  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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>  &amp; <font color="Maroon" title="c15">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>  &amp; <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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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>) )</span><br/><span class="kw">thus </span><a NAME="E14:54_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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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>  &amp; <font color="Maroon" title="c15">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>  &amp; <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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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>) )</span><br/><span class="kw">thus </span><a NAME="E15:54_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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:54_1_2_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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>  &amp; <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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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>) )</span><br/><span class="kw">thus </span><a NAME="E16:54_1_2_2"/>
<font color="Maroon" title="c15">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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>)</span><br/><span class="kw">thus </span><a NAME="E17:54_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="c12">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:54"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E5:54"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<font color="Maroon" title="c4">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="dickson.html#T26" target="_self" title="DICKSON:th.26">Th27</a></span>;<br/>
<span class="kw">set </span><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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> ;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:54"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:54"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
 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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E39">A2</font></span></a>);<br/></span>
<a NAME="E10:54"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:54"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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"><span class="kw">now </span></a><div class="add"><a NAME="E1:54_2"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c5">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E41">A13</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:54_2"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
 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#K2" title="RELSET_1:func.2">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">set </span><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#K2" title="RELSET_1:func.2">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"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<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#K2" title="RELSET_1:func.2">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> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K2" title="RELSET_1:func.2">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>  implies <font color="Maroon" title="c10">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/><span class="kw">assume </span><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#K2" title="RELSET_1:func.2">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><a NAME="E2:54_2_1"/><span class="kw">then </span>
 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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <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="Olive" title="b1">b</font>) &amp; <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> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_2_1"/>
<font color="Maroon" title="c10">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:54_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A21</font></span>: 
 <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#K2" title="RELSET_1:func.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="finset_1.html#T7" title="FINSET_1:th.7">FINSET_1:7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies <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#K2" title="RELSET_1:func.2">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>  )</span><br/><span class="kw">assume </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="RELSET_1:func.2">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> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">x9</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#K1" title="RELSET_1:func.1">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:54_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">s</font>
 ;<br/><a NAME="E5:54_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:54_2_2"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:54_2_2"/><span class="lab"><font color="Green" title="E52">A24</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E5:54_2_2"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_2_2"><span class="lab"><font color="Green" title="E51">A23</font></span></a></span>;<br/><a NAME="E12:54_2_2"/><span class="lab"><font color="Green" title="E53">A25</font></span>: 
<font color="Maroon" title="c11">x9</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_2_2"><span class="lab"><font color="Green" title="E52">A24</font></span></a></span>;<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#K2" title="RELSET_1:func.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_2_2"><span class="lab"><font color="Green" title="E51">A23</font></span></a></span>;<br/><span class="kw">hence </span><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#K2" title="RELSET_1:func.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:54_2_2"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:54_2"/><span class="kw">then </span>
 <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#K2" title="RELSET_1:func.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:54_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_2"><span class="lab"><font color="Green" title="E49">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font><span class="kw"> such that </span><br/><a NAME="E14:54"/><span class="lab"><font color="Green" title="E46">A26</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E15:54"/><span class="lab"><font color="Green" title="E47">A27</font></span>: 
 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="E16:54"/><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:54"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="txt" href="#E15:54"><span class="lab"><font color="Green" title="E47">A27</font></span></a></span>;<br/>
<a NAME="E17:54"/><span class="kw">then </span>
 <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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 
;<br/>
<a NAME="E18:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A29</font></span>: 
 ex <font color="Olive" title="b1">bt</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">b</font> where <font color="Olive" title="b2">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="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">b9</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="Olive" title="b1">bt</font> &amp;  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bt</font> &amp;  not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) )
 
;<br/>
<a NAME="E19:54"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T12" title="NORMSP_1:th.12">NORMSP_1:12</a></span>;<br/>
<a NAME="E20:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A30</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E18:54"><span class="lab"><font color="Green" title="E49">A29</font></span></a>, <a class="ref" href="dickson.html#D11" target="_self" title="DICKSON:def.11">Def11</a></span>;<br/>
<a NAME="E21:54"/>
 <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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</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">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:54"><span class="lab"><font color="Green" title="E48">A28</font></span></a></span>;<br/>
<a NAME="E22:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A31</font></span>: 
 ex <font color="Olive" title="b1">eb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">b</font> where <font color="Olive" title="b2">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="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">b9</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="Olive" title="b1">eb</font> &amp;  ex <font color="Olive" title="b2">eb9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">eb9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">eb</font> &amp;  not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">eb9</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) )
 
;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span><font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> $1;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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/>
<span class="kw">set </span><font color="Maroon" title="c12">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="nat_1.html#K6" title="NAT_1:func.6">.</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="E23:54"/><span class="lab"><font color="Green" title="E52">A32</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S6" title="FINSEQ_1:sch.6">FINSEQ_1:sch 6</a>();<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> </span> } </span>  implies <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>  ) &amp; ( <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> </span> } </span>  ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> </span> } </span>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:54_3_1"/>
<font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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> <span class="kw"> such that </span><br/><a NAME="E3:54_3_1"/><span class="lab"><font color="Green" title="E53">A33</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w</font>
 <span class="kw">and </span><br/><a NAME="E4:54_3_1"/><span class="lab"><font color="Green" title="E54">A34</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c14">w</font>
 ;<br/><a NAME="E5:54_3_1"/><span class="lab"><font color="Green" title="E55">A35</font></span>: 
 <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="c14">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E6:54_3_1"/>
( <font color="Maroon" title="c14">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> or <font color="Maroon" title="c14">w</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
 ;<br/><a NAME="E7:54_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_3_1"><span class="lab"><font color="Green" title="E53">A33</font></span></a>, <a class="txt" href="#E4:54_3_1"><span class="lab"><font color="Green" title="E54">A34</font></span></a>, <a class="txt" href="#E5:54_3_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:54_3_1"/>
<font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:54_3"/><span class="lab"><font color="Green" title="E53">A36</font></span>: 
<font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> </span> } </span> </span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_3_2"/>
( <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_3"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:54_3_2_1"/>
<font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> </span> } </span> </span><br/><div class="add"><a NAME="E2:54_3_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_3_2_1"/>
<font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:54_3_2_2"/>
<font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> </span> } </span> </span><br/><div class="add"><a NAME="E2:54_3_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_3_2_2"/>
<font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E25:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A37</font></span>: 
 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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:54"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E22:54"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E23:54"><span class="lab"><font color="Green" title="E52">A32</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span><font color="Maroon">S<sub>1</sub></font>[$1,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="E26:54"/><span class="lab"><font color="Green" title="E54">A38</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E25:54"><span class="lab"><font color="Green" title="E53">A37</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E55">A39</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">k</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:54_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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="c15">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><span class="kw"> such that </span><br/><a NAME="E3:54_4"/><span class="lab"><font color="Green" title="E56">A40</font></span>: 
<font color="Maroon" title="c14">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E4:54_4"/><span class="lab"><font color="Green" title="E57">A41</font></span>: 
 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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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> 
 <span class="kw">and </span><br/><a NAME="E5:54_4"/><span class="lab"><font color="Green" title="E58">A42</font></span>: 
<font color="Maroon" title="c15">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="c14">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> 
 <span class="kw">and </span><br/><a NAME="E6:54_4"/><span class="lab"><font color="Green" title="E59">A43</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font></span>)</span>)
 ;<br/><span class="kw">set </span><font color="Maroon" title="c16">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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c17">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="c14">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/><span class="kw">set </span><font color="Maroon" title="c18">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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/><span class="kw">consider </span><font color="Maroon" title="c19">BD</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:54_4"/><span class="lab"><font color="Green" title="E60">A44</font></span>: 
<font color="Maroon" title="c19">BD</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c14">N</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E9:54_4"/><span class="lab"><font color="Green" title="E61">A45</font></span>: 
<font color="Maroon" title="c19">BD</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></span>;<br/><a NAME="E10:54_4"/>
<font color="Maroon" title="c19">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="c14">N</font>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E8:54_4"><span class="lab"><font color="Green" title="E60">A44</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/><a NAME="E11:54_4"/><span class="kw">then </span>
<font color="Maroon" title="c19">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="c14">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_4"><span class="lab"><font color="Green" title="E61">A45</font></span></a></span>;<br/><a NAME="E12:54_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c14">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_4"><span class="lab"><font color="Green" title="E58">A42</font></span></a></span>;<br/><a NAME="E13:54_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A46</font></span>: 
 ex <font color="Olive" title="b1">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="c14">N</font>,<font color="Maroon" title="c1">R</font>) st <br/>( <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">BB</font> &amp; <font color="Olive" title="b1">BB</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 ;<br/><a NAME="E14:54_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A47</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c14">N</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">deffunc </span><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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>) <span class="kw">-&gt; </span>   <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>);<br/><a NAME="E1:54_4_1"/><span class="lab"><font color="Green" title="E64">A48</font></span>: 
<font color="Maroon" title="c15">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_4"><span class="lab"><font color="Green" title="E62">A46</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:54_4_1"/><span class="lab"><font color="Green" title="E65">A49</font></span>: 
 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#K2" title="RELSET_1:func.2">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="c15">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">set </span><font color="Maroon" title="c20">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span> ;<br/><a NAME="E3:54_4_1"/><span class="lab"><font color="Green" title="E66">A50</font></span>: 
<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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="#E1:54_4_1"><span class="lab"><font color="Green" title="E64">A48</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span>  implies <font color="Maroon" title="c21">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_4_1_1"/>
<font color="Maroon" title="c21">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><a NAME="E2:54_4_1_1"/><span class="kw">then </span>
 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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Maroon" title="c21">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_4_1_1"/>
<font color="Maroon" title="c21">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_4_1"><span class="lab"><font color="Green" title="E65">A49</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:54_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A51</font></span>: 
 <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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4_1"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="ref" href="finset_1.html#T7" title="FINSET_1:th.7">FINSET_1:7</a></span>;<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c21">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <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="c21">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_4_1_2"/>
<font color="Maroon" title="c21">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">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> <span class="kw"> such that </span><br/><a NAME="E3:54_4_1_2"/><span class="lab"><font color="Green" title="E68">A52</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">w</font>
 <span class="kw">and </span><br/><a NAME="E4:54_4_1_2"/><span class="lab"><font color="Green" title="E69">A53</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c22">w</font>
 <span class="kw">and </span><br/><a NAME="E5:54_4_1_2"/><span class="lab"><font color="Green" title="E70">A54</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">w</font>
 ;<br/>
<a NAME="E6:54_4_1_2"/><span class="lab"><font color="Green" title="E71">A55</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c22">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="txt" href="#E4:54_4_1_2"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="txt" href="#E5:54_4_1_2"><span class="lab"><font color="Green" title="E70">A54</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">sw</font> = <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c22">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/>
<span class="kw">consider </span><font color="Maroon" title="c24">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E9:54_4_1_2"/><span class="lab"><font color="Green" title="E72">A56</font></span>: 
<font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 <span class="kw">and </span><br/><a NAME="E10:54_4_1_2"/><span class="lab"><font color="Green" title="E73">A57</font></span>: 
<font color="Maroon" title="c24">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">sw</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:54_4"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="txt" href="#E6:54_4_1_2"><span class="lab"><font color="Green" title="E71">A55</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/>
<a NAME="E11:54_4_1_2"/><span class="lab"><font color="Green" title="E74">A58</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:54_4"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/>
<a NAME="E12:54_4_1_2"/>
<font color="Maroon" title="c14">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">N</font> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_4_1_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font></span><br/>

<a NAME="E2:54_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A59</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">u</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a></span>;<br/>
<a NAME="E3:54_4_1_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:54_4_1_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_4_1_2_1"><span class="lab"><font color="Green" title="E75">A59</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:54_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:54_4_1_2"><span class="lab"><font color="Green" title="E74">A58</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">b</font> = <font color="Maroon" title="c24">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_4_1_2"><span class="lab"><font color="Green" title="E72">A56</font></span></a></span>;<br/>
<a NAME="E15:54_4_1_2"/><span class="lab"><font color="Green" title="E75">A60</font></span>: 
<font color="Maroon" title="c22">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="c25">b</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_4_1_2"><span class="lab"><font color="Green" title="E70">A54</font></span></a>, <a class="txt" href="#E10:54_4_1_2"><span class="lab"><font color="Green" title="E73">A57</font></span></a></span>;<br/>
<a NAME="E16:54_4_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c25">b</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_4_1_2"><span class="lab"><font color="Green" title="E72">A56</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:54_4_1_2"/>
<font color="Maroon" title="c21">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#K2" title="RELSET_1:func.2">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="c15">B</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4_1_2"><span class="lab"><font color="Green" title="E68">A52</font></span></a>, <a class="txt" href="#E15:54_4_1_2"><span class="lab"><font color="Green" title="E75">A60</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:54_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_4"><span class="lab"><font color="Green" title="E57">A41</font></span></a>, <a class="txt" href="#E5:54_4_1"><span class="lab"><font color="Green" title="E67">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font><span class="kw"> such that </span><br/><a NAME="E17:54_4"/><span class="lab"><font color="Green" title="E64">A61</font></span>: 
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 <span class="kw">and </span><br/><a NAME="E18:54_4"/><span class="lab"><font color="Green" title="E65">A62</font></span>: 
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">b</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/><a NAME="E19:54_4"/>
<font color="Maroon" title="c20">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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:54_4"><span class="lab"><font color="Green" title="E64">A61</font></span></a>, <a class="txt" href="#E18:54_4"><span class="lab"><font color="Green" title="E65">A62</font></span></a></span>;<br/><a NAME="E20:54_4"/><span class="kw">then </span>
 <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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c15">B</font><span class="kw"> such that </span><br/><a NAME="E22:54_4"/><span class="lab"><font color="Green" title="E66">A63</font></span>: 
<font color="Maroon" title="c21">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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E23:54_4"/><span class="lab"><font color="Green" title="E67">A64</font></span>: 
 ex <font color="Olive" title="b1">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b</font> &amp;  not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 ;<br/><a NAME="E24:54_4"/><span class="lab"><font color="Green" title="E68">A65</font></span>: 
<font color="Maroon" title="c21">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 ;<br/><a NAME="E25:54_4"/>
<font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:54_4"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/><a NAME="E26:54_4"/><span class="kw">then </span>
<font color="Maroon" title="c21">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:54_4"><span class="lab"><font color="Green" title="E68">A65</font></span></a></span>;<br/><a NAME="E27:54_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A66</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c21">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a></span>;<br/><a NAME="E28:54_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A67</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c15">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_4"><span class="lab"><font color="Green" title="E59">A43</font></span></a>, <a class="txt" href="#E22:54_4"><span class="lab"><font color="Green" title="E66">A63</font></span></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></span>;<br/><a NAME="E29:54_4"/><span class="lab"><font color="Green" title="E71">A68</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_4"><span class="lab"><font color="Green" title="E59">A43</font></span></a>, <a class="txt" href="#E22:54_4"><span class="lab"><font color="Green" title="E66">A63</font></span></a>, <a class="txt" href="#E27:54_4"><span class="lab"><font color="Green" title="E69">A66</font></span></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/><span class="kw">defpred </span><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> ] <span class="kw">means </span><font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> $1;<br/><span class="kw">set </span><font color="Maroon" title="c22">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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/><span class="kw">set </span><font color="Maroon" title="c23">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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/><span class="kw">set </span><font color="Maroon" title="c24">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="E30:54_4"/><span class="lab"><font color="Green" title="E72">A69</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="dickson.html#S1" target="_self" title="DICKSON:sch.1">DICKSON:sch 1</a>();<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span> } </span>  implies <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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>  ) &amp; ( <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) </span> } </span>  ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) </span> } </span>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:54_4_2_1"/>
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">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> <span class="kw"> such that </span><br/><a NAME="E3:54_4_2_1"/><span class="lab"><font color="Green" title="E73">A70</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E4:54_4_2_1"/><span class="lab"><font color="Green" title="E74">A71</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E5:54_4_2_1"/><span class="lab"><font color="Green" title="E75">A72</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">w</font>
 ;<br/><a NAME="E6:54_4_2_1"/>
( <font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ;<br/><a NAME="E7:54_4_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c25">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4_2_1"><span class="lab"><font color="Green" title="E73">A70</font></span></a>, <a class="txt" href="#E4:54_4_2_1"><span class="lab"><font color="Green" title="E74">A71</font></span></a>, <a class="txt" href="#E5:54_4_2_1"><span class="lab"><font color="Green" title="E75">A72</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:54_4_2_1"/>
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:54_4_2"/><span class="lab"><font color="Green" title="E73">A73</font></span>: 
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) </span> } </span> </span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_4_2_2"/>
( <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c25">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_4_2"><span class="lab"><font color="Green" title="E73">A73</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:54_4_2_2_1"/>
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) </span> } </span> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">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> <span class="kw"> such that </span><br/><a NAME="E3:54_4_2_2_1"/><span class="lab"><font color="Green" title="E74">A74</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E4:54_4_2_2_1"/><span class="lab"><font color="Green" title="E75">A75</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E5:54_4_2_2_1"/><span class="lab"><font color="Green" title="E76">A76</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c26">w</font>
 ;<br/><a NAME="E6:54_4_2_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:54_4"><span class="lab"><font color="Green" title="E71">A68</font></span></a>, <a class="txt" href="#E5:54_4_2_2_1"><span class="lab"><font color="Green" title="E76">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:54_4_2_2_1"/>
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_4_2_2_1"><span class="lab"><font color="Green" title="E74">A74</font></span></a>, <a class="txt" href="#E4:54_4_2_2_1"><span class="lab"><font color="Green" title="E75">A75</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:54_4_2_2_2"/>
<font color="Maroon" title="c25">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive">b<sub>2</sub></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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) </span> } </span> </span><br/><div class="add"><a NAME="E2:54_4_2_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_4_2_2_2"/>
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E32:54_4"/><span class="kw">then </span>
 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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:54_4"><span class="lab"><font color="Green" title="E66">A63</font></span></a>, <a class="txt" href="#E23:54_4"><span class="lab"><font color="Green" title="E67">A64</font></span></a>, <a class="txt" href="#E28:54_4"><span class="lab"><font color="Green" title="E70">A67</font></span></a>, <a class="txt" href="#E30:54_4"><span class="lab"><font color="Green" title="E72">A69</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><span class="kw">hence </span><a NAME="E33:54_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E43">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E28:54"/><span class="lab"><font color="Green" title="E56">A77</font></span>: 
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E26:54"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E27:54"><span class="lab"><font color="Green" title="E55">A39</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c13">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/>
<span class="kw">take </span>
<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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c2">s</font> &amp; <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="dickson.html#V2" title="DICKSON:attr.2">weakly-ascending</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><a NAME="E1:54_5_1"/>
<font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">fn</font> = <font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E45">A17</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">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="c18">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><span class="kw"> such that </span><br/><a NAME="E4:54_5_1"/><span class="lab"><font color="Green" title="E57">A78</font></span>: 
<font color="Maroon" title="c17">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c16">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E5:54_5_1"/><span class="lab"><font color="Green" title="E58">A79</font></span>: 
 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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c16">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> 
 <span class="kw">and </span><br/><a NAME="E6:54_5_1"/><span class="lab"><font color="Green" title="E59">A80</font></span>: 
<font color="Maroon" title="c18">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="c17">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> 
 <span class="kw">and </span><br/><a NAME="E7:54_5_1"/><span class="lab"><font color="Green" title="E60">A81</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">fn</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> </span>)</span>,<font color="Maroon" title="c16">fn</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:54"><span class="lab"><font color="Green" title="E56">A77</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">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="c17">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/><span class="kw">set </span><font color="Maroon" title="c20">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">fn</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c21">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">fn</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c22">BD</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:54_5_1"/><span class="lab"><font color="Green" title="E61">A82</font></span>: 
<font color="Maroon" title="c22">BD</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E10:54_5_1"/><span class="lab"><font color="Green" title="E62">A83</font></span>: 
<font color="Maroon" title="c22">BD</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></span>;<br/><a NAME="E11:54_5_1"/>
<font color="Maroon" title="c22">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="c17">N</font>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E9:54_5_1"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/><a NAME="E12:54_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">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="c17">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54_5_1"><span class="lab"><font color="Green" title="E62">A83</font></span></a></span>;<br/><a NAME="E13:54_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">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="c17">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_5_1"><span class="lab"><font color="Green" title="E59">A80</font></span></a></span>;<br/><a NAME="E14:54_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A84</font></span>: 
 ex <font color="Olive" title="b1">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="c17">N</font>,<font color="Maroon" title="c1">R</font>) st <br/>( <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">BB</font> &amp; <font color="Olive" title="b1">BB</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 ;<br/><a NAME="E15:54_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A85</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/><a NAME="E16:54_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A86</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:54_5_1_1"/><span class="lab"><font color="Green" title="E66">A87</font></span>: 
<font color="Maroon" title="c18">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:54_5_1"><span class="lab"><font color="Green" title="E63">A84</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:54_5_1_1"/><span class="lab"><font color="Green" title="E67">A88</font></span>: 
 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#K2" title="RELSET_1:func.2">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="c18">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c16">fn</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">deffunc </span><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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">H<sub>1</sub></font>($1,<font color="Maroon" title="c16">fn</font>);<br/><span class="kw">set </span><font color="Maroon" title="c23">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c24">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c16">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"/><span class="lab"><font color="Green" title="E68">A89</font></span>: 
<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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="#E1:54_5_1_1"><span class="lab"><font color="Green" title="E66">A87</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  implies <font color="Maroon" title="c25">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_5_1_1_1"/>
<font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><a NAME="E2:54_5_1_1_1"/><span class="kw">then </span>
 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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c16">fn</font>) &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_5_1_1_1"/>
<font color="Maroon" title="c25">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_5_1_1"><span class="lab"><font color="Green" title="E67">A88</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:54_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A90</font></span>: 
 <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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_5_1_1"><span class="lab"><font color="Green" title="E68">A89</font></span></a>, <a class="ref" href="finset_1.html#T7" title="FINSET_1:th.7">FINSET_1:7</a></span>;<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c16">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c16">fn</font> <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="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_5_1_1_2"/>
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c16">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">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> <span class="kw"> such that </span><br/><a NAME="E3:54_5_1_1_2"/><span class="lab"><font color="Green" title="E70">A91</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E4:54_5_1_1_2"/><span class="lab"><font color="Green" title="E71">A92</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E5:54_5_1_1_2"/><span class="lab"><font color="Green" title="E72">A93</font></span>: 
<font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">w</font>
 ;<br/>
<a NAME="E6:54_5_1_1_2"/><span class="lab"><font color="Green" title="E73">A94</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_5_1"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="txt" href="#E4:54_5_1_1_2"><span class="lab"><font color="Green" title="E71">A92</font></span></a>, <a class="txt" href="#E5:54_5_1_1_2"><span class="lab"><font color="Green" title="E72">A93</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">sw</font> = <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">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/>
<span class="kw">consider </span><font color="Maroon" title="c28">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E9:54_5_1_1_2"/><span class="lab"><font color="Green" title="E74">A95</font></span>: 
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">and </span><br/><a NAME="E10:54_5_1_1_2"/><span class="lab"><font color="Green" title="E75">A96</font></span>: 
<font color="Maroon" title="c28">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">sw</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:54_5_1"><span class="lab"><font color="Green" title="E64">A85</font></span></a>, <a class="txt" href="#E6:54_5_1_1_2"><span class="lab"><font color="Green" title="E73">A94</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/>
<a NAME="E11:54_5_1_1_2"/><span class="lab"><font color="Green" title="E76">A97</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:54_5_1"><span class="lab"><font color="Green" title="E64">A85</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/>
<a NAME="E12:54_5_1_1_2"/>
<font color="Maroon" title="c17">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font> or <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_5_1_1_2_1"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font></span><br/>

<a NAME="E2:54_5_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A98</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c16">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">u</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_5_1"><span class="lab"><font color="Green" title="E57">A78</font></span></a></span>;<br/>
<a NAME="E3:54_5_1_1_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:54_5_1_1_2_1"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_5_1_1_2_1"><span class="lab"><font color="Green" title="E77">A98</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:54_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:54_5_1_1_2"><span class="lab"><font color="Green" title="E76">A97</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">b</font> = <font color="Maroon" title="c28">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_5_1_1_2"><span class="lab"><font color="Green" title="E74">A95</font></span></a></span>;<br/>
<a NAME="E15:54_5_1_1_2"/><span class="lab"><font color="Green" title="E77">A99</font></span>: 
<font color="Maroon" title="c26">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="c29">b</font>,<font color="Maroon" title="c16">fn</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_5_1_1_2"><span class="lab"><font color="Green" title="E72">A93</font></span></a>, <a class="txt" href="#E10:54_5_1_1_2"><span class="lab"><font color="Green" title="E75">A96</font></span></a></span>;<br/>
<a NAME="E16:54_5_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c29">b</font>,<font color="Maroon" title="c16">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_5_1_1_2"><span class="lab"><font color="Green" title="E74">A95</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:54_5_1_1_2"/>
<font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_5_1_1_2"><span class="lab"><font color="Green" title="E70">A91</font></span></a>, <a class="txt" href="#E15:54_5_1_1_2"><span class="lab"><font color="Green" title="E77">A99</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:54_5_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_5_1"><span class="lab"><font color="Green" title="E58">A79</font></span></a>, <a class="txt" href="#E5:54_5_1_1"><span class="lab"><font color="Green" title="E69">A90</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font><span class="kw"> such that </span><br/><a NAME="E19:54_5_1"/><span class="lab"><font color="Green" title="E66">A100</font></span>: 
<font color="Maroon" title="c23">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">and </span><br/><a NAME="E20:54_5_1"/><span class="lab"><font color="Green" title="E67">A101</font></span>: 
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c16">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="c23">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">fn</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:54_5_1"><span class="lab"><font color="Green" title="E66">A100</font></span></a>, <a class="txt" href="#E20:54_5_1"><span class="lab"><font color="Green" title="E67">A101</font></span></a></span>;<br/><a NAME="E22:54_5_1"/><span class="kw">then </span>
 <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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">fn</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 ;<br/><a NAME="E23:54_5_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">B</font> st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">b</font> where <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">B</font> :  ex <font color="Olive" title="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">b9</font>,<font color="Maroon" title="c16">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="Olive" title="b1">b</font> &amp;  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">fn</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) )
 ;<br/><a NAME="E24:54_5_1"/><span class="kw">then </span>
 <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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c16">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="c17">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:54_5_1"><span class="lab"><font color="Green" title="E65">A86</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E25:54_5_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">b</font> where <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">B</font> :  ex <font color="Olive" title="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">b9</font>,<font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">fn</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c16">fn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_5_1"><span class="lab"><font color="Green" title="E57">A78</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E26:54_5_1"/>
<font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_5_1"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:54_5"/>
<font color="Maroon" title="c14">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#D6" title="SEQM_3:def.6">SEQM_3:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <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> </span><br/><span class="kw">let </span><font color="Maroon" title="c15">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">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> </span><br/><a NAME="E3:54_5"/>
<font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:54_5"/>
<font color="Maroon" title="c14">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E45">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c14">f</font> as   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<a NAME="E32:54"/>
<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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">s</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c15">f</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E33:54"/>
<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="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c2">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="dickson.html#V2" title="DICKSON:attr.2">weakly-ascending</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c16">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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="dickson.html#D2" target="_self" title="DICKSON:def.2">DICKSON:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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></span><br/>

<a NAME="E34:54"/><span class="lab"><font color="Green" title="E57">A102</font></span>: 
<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/>
<a NAME="E35:54"/><span class="lab"><font color="Green" title="E58">A103</font></span>: 
<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">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="c18">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><span class="kw"> such that </span><br/><a NAME="E37:54"/><span class="lab"><font color="Green" title="E59">A104</font></span>: 
<font color="Maroon" title="c17">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E38:54"/><span class="lab"><font color="Green" title="E60">A105</font></span>: 
 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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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> 
 <span class="kw">and </span><br/><a NAME="E39:54"/><span class="lab"><font color="Green" title="E61">A106</font></span>: 
<font color="Maroon" title="c18">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="c17">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> 
 <span class="kw">and </span><br/><a NAME="E40:54"/><span class="lab"><font color="Green" title="E62">A107</font></span>: 
<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:54"><span class="lab"><font color="Green" title="E56">A77</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><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="c17">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/>
<span class="kw">consider </span><font color="Maroon" title="c22">BD</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E42:54"/><span class="lab"><font color="Green" title="E63">A108</font></span>: 
<font color="Maroon" title="c22">BD</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E43:54"/><span class="lab"><font color="Green" title="E64">A109</font></span>: 
<font color="Maroon" title="c22">BD</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></span>;<br/>
<a NAME="E44:54"/>
<font color="Maroon" title="c22">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="c17">N</font>,<font color="Maroon" title="c1">R</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E42:54"><span class="lab"><font color="Green" title="E63">A108</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/>
<a NAME="E45:54"/><span class="kw">then </span>
<font color="Maroon" title="c22">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="c17">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:54"><span class="lab"><font color="Green" title="E64">A109</font></span></a></span>;<br/>
<a NAME="E46:54"/><span class="kw">then </span>
<font color="Maroon" title="c18">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="c17">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:54"><span class="lab"><font color="Green" title="E61">A106</font></span></a></span>;<br/>
<a NAME="E47:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A110</font></span>: 
 ex <font color="Olive" title="b1">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="c17">N</font>,<font color="Maroon" title="c1">R</font>) st <br/>( <font color="Olive" title="b1">BB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">B</font> &amp; <font color="Olive" title="b1">BB</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 
;<br/>
<a NAME="E48:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A111</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="dickson.html#D13" target="_self" title="DICKSON:def.13">Def13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6"><span class="kw">now </span></a><div class="add"><a NAME="E1:54_6"/><span class="lab"><font color="Green" title="E67">A112</font></span>: 
<font color="Maroon" title="c18">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:54"><span class="lab"><font color="Green" title="E65">A110</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:54_6"/><span class="lab"><font color="Green" title="E68">A113</font></span>: 
 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#K2" title="RELSET_1:func.2">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="c18">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">deffunc </span><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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">H<sub>1</sub></font>($1,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>);<br/><span class="kw">set </span><font color="Maroon" title="c23">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c24">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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"/><span class="lab"><font color="Green" title="E69">A114</font></span>: 
<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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="#E1:54_6"><span class="lab"><font color="Green" title="E67">A112</font></span></a>);<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  implies <font color="Maroon" title="c25">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_6_1"/>
<font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><a NAME="E2:54_6_1"/><span class="kw">then </span>
 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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:54_6_1"/>
<font color="Maroon" title="c25">X</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_6"><span class="lab"><font color="Green" title="E68">A113</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:54_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A115</font></span>: 
 <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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_6"><span class="lab"><font color="Green" title="E69">A114</font></span></a>, <a class="ref" href="finset_1.html#T7" title="FINSET_1:th.7">FINSET_1:7</a></span>;<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <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="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_6_2"/>
<font color="Maroon" title="c25">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">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> <span class="kw"> such that </span><br/><a NAME="E3:54_6_2"/><span class="lab"><font color="Green" title="E71">A116</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E4:54_6_2"/><span class="lab"><font color="Green" title="E72">A117</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">and </span><br/><a NAME="E5:54_6_2"/><span class="lab"><font color="Green" title="E73">A118</font></span>: 
<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">w</font>
 ;<br/>
<a NAME="E6:54_6_2"/><span class="lab"><font color="Green" title="E74">A119</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:54"><span class="lab"><font color="Green" title="E59">A104</font></span></a>, <a class="txt" href="#E4:54_6_2"><span class="lab"><font color="Green" title="E72">A117</font></span></a>, <a class="txt" href="#E5:54_6_2"><span class="lab"><font color="Green" title="E73">A118</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">sw</font> = <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c26">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/>
<span class="kw">consider </span><font color="Maroon" title="c28">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E9:54_6_2"/><span class="lab"><font color="Green" title="E75">A120</font></span>: 
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">and </span><br/><a NAME="E10:54_6_2"/><span class="lab"><font color="Green" title="E76">A121</font></span>: 
<font color="Maroon" title="c28">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">sw</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:54"><span class="lab"><font color="Green" title="E66">A111</font></span></a>, <a class="txt" href="#E6:54_6_2"><span class="lab"><font color="Green" title="E74">A119</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/>
<a NAME="E11:54_6_2"/><span class="lab"><font color="Green" title="E77">A122</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:54"><span class="lab"><font color="Green" title="E66">A111</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/>
<a NAME="E12:54_6_2"/>
<font color="Maroon" title="c17">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font> or <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_6_2_1"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font></span><br/>

<a NAME="E2:54_6_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A123</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">u</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:54"><span class="lab"><font color="Green" title="E59">A104</font></span></a></span>;<br/>
<a NAME="E3:54_6_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:54_6_2_1"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_6_2_1"><span class="lab"><font color="Green" title="E78">A123</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:54_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:54_6_2"><span class="lab"><font color="Green" title="E77">A122</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">b</font> = <font color="Maroon" title="c28">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_6_2"><span class="lab"><font color="Green" title="E75">A120</font></span></a></span>;<br/>
<a NAME="E15:54_6_2"/><span class="lab"><font color="Green" title="E78">A124</font></span>: 
<font color="Maroon" title="c26">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="c29">b</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_6_2"><span class="lab"><font color="Green" title="E73">A118</font></span></a>, <a class="txt" href="#E10:54_6_2"><span class="lab"><font color="Green" title="E76">A121</font></span></a></span>;<br/>
<a NAME="E16:54_6_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c29">b</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_6_2"><span class="lab"><font color="Green" title="E75">A120</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:54_6_2"/>
<font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">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="c18">B</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_6_2"><span class="lab"><font color="Green" title="E71">A116</font></span></a>, <a class="txt" href="#E15:54_6_2"><span class="lab"><font color="Green" title="E78">A124</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:54_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:54"><span class="lab"><font color="Green" title="E60">A105</font></span></a>, <a class="txt" href="#E5:54_6"><span class="lab"><font color="Green" title="E70">A115</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font><span class="kw"> such that </span><br/><a NAME="E51:54"/><span class="lab"><font color="Green" title="E67">A125</font></span>: 
<font color="Maroon" title="c23">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">and </span><br/><a NAME="E52:54"/><span class="lab"><font color="Green" title="E68">A126</font></span>: 
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/>
<a NAME="E53:54"/>
<font color="Maroon" title="c23">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:54"><span class="lab"><font color="Green" title="E67">A125</font></span></a>, <a class="txt" href="#E52:54"><span class="lab"><font color="Green" title="E68">A126</font></span></a></span>;<br/>
<a NAME="E54:54"/><span class="kw">then </span>
 <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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span> 
 
;<br/>
<a NAME="E55:54"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">B</font> st <br/>( <font color="Olive" title="b1">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="b2">b</font> where <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">B</font> :  ex <font color="Olive" title="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) </span> } </span>  &amp;  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) )
 
;<br/>
<a NAME="E56:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A127</font></span>: 
 <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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="c18">B</font>
 
;<br/>
<a NAME="E57:54"/>
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:54"><span class="lab"><font color="Green" title="E66">A111</font></span></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></span>;<br/>
<a NAME="E58:54"/><span class="kw">then </span>
 <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="c18">B</font> :  ex <font color="Olive" title="b2">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b2">b9</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">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="c17">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:54"><span class="lab"><font color="Green" title="E69">A127</font></span></a></span>;<br/>
<a NAME="E59:54"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">b</font> where <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">B</font> :  ex <font color="Olive" title="b3">b9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> st <br/>( <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">b9</font>,<font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c2">s</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c15">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:54"><span class="lab"><font color="Green" title="E59">A104</font></span></a></span>;<br/>
<a NAME="E60:54"/><span class="kw">then </span>
<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:54"><span class="lab"><font color="Green" title="E57">A102</font></span></a>, <a class="txt" href="#E35:54"><span class="lab"><font color="Green" title="E58">A103</font></span></a>, <a class="txt" href="#E40:54"><span class="lab"><font color="Green" title="E62">A107</font></span></a>, <a class="ref" href="dickson.html#D12" target="_self" title="DICKSON:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E61:54"/>
<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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c16">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="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
