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

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> ;<br/>
<a NAME="E1:39_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_1"/>
(  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:39_1_1_1_1_1_1"/><b>then </b>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E3:39_1_1_1_1_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E4:39_1_1_1_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1_2"/><i><font color="Green">E28</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:39_1_1_1_1_1_2"/><i><font color="Green">E29</font></i>: 
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T36">QC_LANG3:36</a></i>;<br/><a NAME="E6:39_1_1_1_1_1_2"/><i><font color="Green">E33</font></i>: 
 ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:39_1_1_1_1_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E7:39_1_1_1_1_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T1" target="_self">Th1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/> <a href="qc_lang3.html#K2">x.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E8:39_1_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">b</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1_1_1_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> ;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:39_1_1_1_1_1_2_2"/><i><font color="Green">E40</font></i>: 
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T36">QC_LANG3:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E6:39_1_1_1_1_1_2_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">i</font>] )
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:39_1_1_1_1_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>;<br/><a NAME="E9:39_1_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S10">FRAENKEL:sch 10</a>();<br/></i><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:39_1_1_1_1_1_2"/><i><font color="Green">E69</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>and </b><br/><a NAME="E12:39_1_1_1_1_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>, <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="cqc_sim1.html#E5:39_1_1_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:39_1_1_1_1_1_2_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_1_1_1_2_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1_1_1_2_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:39_1_1_1_1_1_2_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E6:39_1_1_1_1_1_2_3_1"/>
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E5:39_1_1_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E7:39_1_1_1_1_1_2_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:39_1_1_1_1_1_2_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:39_1_1_1_1_1_2_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> =  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">f</font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E7:39_1_1_1_1_1_2_3"/>
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E8:39_1_1_1_1_1_2_3"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E9:39_1_1_1_1_1_2_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E5:39_1_1_1_1_1_2"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E15:39_1_1_1_1_1_2"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E16:39_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, , <a class="txt" href="cqc_sim1.html#E1:39_1_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E17:39_1_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">l</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>,<font color="Olive">k</font>] )
 ;<br/><a NAME="E18:39_1_1_1_1_1_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">l</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">l</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">l</font>,<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">m</font>]
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:39_1_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 <b>and </b><br/><a NAME="E21:39_1_1_1_1_1_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span>  holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">m</font>,<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="cqc_sim1.html#S4" target="_self">CQC_SIM1:sch 4</a>(, , <a class="txt" href="cqc_sim1.html#E6:39_1_1_1_1_1_2"><i><font color="Green">E33</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_2_4"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">n</font> = <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>thus </b><a NAME="E1:39_1_1_1_1_1_2_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E2:39_1_1_1_1_1_2_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:39_1_1_1_1_1_2_4"/><i><font color="Green">E81</font></i>: 
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E4:39_1_1_1_1_1_2_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E5:39_1_1_1_1_1_2_4"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E6:39_1_1_1_1_1_2_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E23:39_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E24:39_1_1_1_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1"/>
not  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">p</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div>
