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

<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> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>] &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">a<sub>4</sub></font> );<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E2:11_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>and </b><br/><a NAME="E3:11_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">c</font>,<font color="Olive">n</font>) where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">fq</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">V</font>,<font color="Olive">n</font>,<font color="Olive">fq</font>] </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">V</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="pcomps_2.html#S2" target="_self">PCOMPS_2:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E4:11_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i/>;<br/>

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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">c</font>,<font color="Maroon">n</font>) where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : for <font color="Olive">fq</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> holds <br/>( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">n</font>] &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">fq</font> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">c</font>,<font color="Maroon">n</font>) where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">n</font>] &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">fxxx1</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">V</font>) where <font color="Olive">V</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">V</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">fxxx</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">V</font>) where <font color="Olive">V</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">V</font>] </span>}</span> ;<br/>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">V</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E1:11_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">V</font>]
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">n</font>);<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">fq</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> holds <br/>( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">V</font>,<font color="Maroon">n</font>] &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">fq</font> );<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">V</font>,<font color="Maroon">n</font>] &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q1</font></span>)</span></span>)</span> where <font color="Olive">q1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  );<br/><div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><a NAME="E1:11_1_1_1"/><i><font color="Green">E27</font></i>: 
( ( for <font color="Olive">fq</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> holds <br/> not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">fq</font> ) iff not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_1_1_1_1"/>
( ( for <font color="Olive">fq</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> holds <br/> not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">fq</font> ) implies  not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1_1_1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">fq</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> holds <br/> not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">fq</font>
 ;<br/>

<b>assume </b><a NAME="E2:11_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:11_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:11_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">q</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="E7:11_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E8:11_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:11_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:11_1_1_1_1"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">fq</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">q</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="E3:11_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:11_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 ;<br/>

<b>assume </b><a NAME="E5:11_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">fq</font>
 ;<br/>

<a NAME="E6:11_1_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">fq</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font></span>)</span></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:11_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:11_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c</font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_2"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">fq</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> holds <br/>( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c</font>,<font color="Maroon">V</font>,<font color="Maroon">n</font>] &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">fq</font> )
 ;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:11_1_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><b>consider </b><font color="Maroon">fq</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:11_1_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 ;<br/><b>thus </b><a NAME="E6:11_1_1_1_2_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c</font>,<font color="Maroon">V</font>,<font color="Maroon">n</font>]
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E7:11_1_1_1_2_1"/>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font></span>)</span></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:11_1_1_1_2"/>
( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c</font>,<font color="Maroon">V</font>,<font color="Maroon">n</font>] &amp; not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  )
 ;<br/>

<b>hence </b><a NAME="E3:11_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:11_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>();<br/></i><b>hence </b><a NAME="E4:11_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">V</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">V</font>)
 ;<br/></div><b>end;</b></div>
<a NAME="E6:11_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">V</font>) where <font color="Olive">V</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">V</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">V</font>) where <font color="Olive">V</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">V</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S6">FRAENKEL:sch 6</a>();<br/></i>
<b>hence </b><a NAME="E7:11_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">c</font>,<font color="Maroon">n</font>) where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Olive">V</font>,<font color="Maroon">n</font>] &amp; not <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span>  ) </span>}</span> </span>)</span> where <font color="Olive">V</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">V</font>] </span>}</span> 
 <b>by </b><i/>;<br/>


</div>
