<?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> ] <b>means </b> ex <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] ) ) &amp; ( for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">O</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] ) ) holds <br/><font color="Olive">O2</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">O</font> ) );<br/>
<a NAME="E1:1_1"/><i><font color="Green">E37</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> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">y</font>] );<br/>
<a NAME="E1:1_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x'</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x'</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x'</font>,<font color="Olive">o</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">x'</font>,<font color="Olive">y</font>] );<br/>
<a NAME="E2:1_1_1_1"/><i><font color="Green">E42</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="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>thus </b><a NAME="E3:1_1_1_1_1"/>
 ex <font color="Olive">o</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">o</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Olive">o</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">x'</font>,<font color="Olive">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:1_1_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i'</font>,<font color="Maroon">x'</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">o</font> =  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font>;<br/>

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

<b>thus </b><a NAME="E3:1_1_1_1_1_1"/>
<font color="Maroon">o</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 ;<br/>

<a NAME="E4:1_1_1_1_1_1"/>
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T81">CLASSES1:81</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T73">CLASSES1:73</a></i>;<br/>

<b>thus </b><a NAME="E6:1_1_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">x'</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:1_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>take </b><font color="Maroon">o</font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>;<br/>

<b>thus </b><a NAME="E6:1_1_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">o</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">o</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">x'</font>,<font color="Olive">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>and </b><br/><a NAME="E3:1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">x'</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E5:1_1_1_1_2"/>
<font color="Maroon">o</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 ;<br/>

<a NAME="E6:1_1_1_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:1_1_1_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E8:1_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">O</font> = <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i/>;<br/>
<a NAME="E10:1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="classes1.html#K4">Rank</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/>
<a NAME="E11:1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T81">CLASSES1:81</a></i>;<br/>
<a NAME="E12:1_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T81">CLASSES1:81</a></i>;<br/>

<b>thus </b><a NAME="E14:1_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">x'</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">x'</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x'</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x'</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] );<br/>
<a NAME="E5:1_1_1"/><i><font color="Green">E58</font></i>: 
 ex <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">O</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">O2</font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:1_1_1_2"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> for <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="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:1_1_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>and </b><br/><a NAME="E4:1_1_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:1_1_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font>]
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<a NAME="E6:1_1_1_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:1_1_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E8:1_1_1_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O2</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">O</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i/>;<br/>
<a NAME="E10:1_1_1_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="classes1.html#K4">Rank</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O2</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/>
<a NAME="E11:1_1_1_2_1"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T81">CLASSES1:81</a></i>;<br/>
<b>hence </b><a NAME="E12:1_1_1_2_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>thus </b><a NAME="E13:1_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font>]
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">O2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">O2</font>] &amp; ( for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">O</font>] holds <br/><font color="Maroon">O2</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">O</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">O2</font>
;<br/>

<b>thus </b><a NAME="E8:1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">O2</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1_1"/><i><font color="Green">E66</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> <br/> for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">y1</font>]
 <b>and </b><br/><a NAME="E2:1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">y2</font>]
 ;<br/>

<b>consider </b><font color="Maroon">O1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E5:1_1_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O1</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] )
 <b>and </b><br/><a NAME="E6:1_1_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">O</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] ) ) holds <br/><font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">O2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>and </b><br/><a NAME="E9:1_1_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] )
 <b>and </b><br/><a NAME="E10:1_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">O</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] ) ) holds <br/><font color="Maroon">O2</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">O</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:1_1_2"/>
<font color="Maroon">O2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:1_1"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E6:1_1"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S1">RECDEF_1:sch 1</a>(, );<br/></i>
<b>set </b><font color="Maroon">beta</font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ,<font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a> ] &amp; <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> ,<font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> ,<font color="Olive">y</font>] or  not  <a href="classes1.html#K6">the_rank_of</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> ) ) );<br/>
<a NAME="E7:1_1"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:1_1_3"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T74">CLASSES1:74</a></i>;<br/>
<b>then consider </b><font color="Maroon">alfa</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:1_1_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">alfa</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E10:1_1_3"/><i><font color="Green">E92</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">alfa</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<a NAME="E11:1_1_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">alfa</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T73">CLASSES1:73</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:1_1_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E14:1_1_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">alfa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E16:1_1_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">alfa</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">O2</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E18:1_1_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> ,<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> ,<font color="Olive">y</font>] );<br/>
<a NAME="E19:1_1_3"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">alfa</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T81">CLASSES1:81</a></i>;<br/>
<a NAME="E20:1_1_3"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> ,<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> ,<font color="Olive">y</font>] )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:1_1_3"/><b>then </b><i><font color="Green">E99</font></i>: 
 ex <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">O</font>]
 
;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E23:1_1_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">O</font>]
 <b>and </b><br/><a NAME="E24:1_1_3"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">O'</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">O'</font>] holds <br/><font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">O'</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:1_1_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E27:1_1_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">Y</font>]
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">u</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>

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

<b>thus </b><a NAME="E28:1_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>

<b>thus </b><a NAME="E29:1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> ,<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> ,<font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a> ]
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>

<b>thus </b><a NAME="E30:1_1_3"/>
<font color="Maroon">u</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>

<b>given </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E31:1_1_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> ,<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> ,<font color="Maroon">y</font>]
 <b>and </b><br/><a NAME="E32:1_1_3"/><i><font color="Green">E106</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>
 ;<br/>

<a NAME="E33:1_1_3"/><i><font color="Green">E107</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E34:1_1_3"/><i><font color="Green">E108</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">Y</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T73">CLASSES1:73</a></i>;<br/>
<a NAME="E35:1_1_3"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/>
<a NAME="E36:1_1_3"/><b>then </b>
<font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E37:1_1_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E10:1_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> ;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:1_1"/><i><font color="Green">E112</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E13:1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E14:1_1"/><i><font color="Green">E114</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_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3">RECDEF_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E15:1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E16:1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><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> ] <b>means </b> <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>;<br/>
<a NAME="E17:1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E19:1_1"/><i><font color="Green">E116</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>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_4"/><i><font color="Green">E117</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:1_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T74">CLASSES1:74</a></i>;<br/>
<a NAME="E3:1_1_4"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>then 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="E6:1_1_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E7:1_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> ]
 <b>and </b><br/><a NAME="E8:1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E9:1_1_4"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Olive">y</font>] or  not  <a href="classes1.html#K6">the_rank_of</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:1_1_4"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">o1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E12:1_1_4"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E13:1_1_4"/><i><font color="Green">E123</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:1_1_4"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E16:1_1_4"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</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">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">o2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E19:1_1_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E20:1_1_4"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] )
 <b>and </b><br/><a NAME="E21:1_1_4"/>
for <font color="Olive">o</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> 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">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">o</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>,<font color="Olive">Y</font>] ) ) holds <br/><font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">o</font>
 <b>by </b><i/>;<br/>
<a NAME="E22:1_1_4"/>
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="classes1.html#T81">CLASSES1:81</a></i>;<br/>
<a NAME="E23:1_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T73">CLASSES1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">YY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E25:1_1_4"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">YY</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">o2</font>
 <b>and </b><br/><a NAME="E26:1_1_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">YY</font>]
 <b>by </b><i/>;<br/>
<a NAME="E27:1_1_4"/>
(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">YY</font>] or  not  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">YY</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E28:1_1_4"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">YY</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<a NAME="E29:1_1_4"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">YY</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T73">CLASSES1:73</a></i>;<br/>
<a NAME="E30:1_1_4"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:1_1_4"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E32:1_1_4"/><b>then </b>
<font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E33:1_1_4"/>
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:1_1"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E21:1_1"/><i><font color="Green">E135</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E22:1_1"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T74">CLASSES1:74</a></i>;<br/>
<a NAME="E24:1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then 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="E26:1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
 <b>and </b><br/><a NAME="E27:1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> ]
 <b>and </b><br/><a NAME="E28:1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E29:1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Olive">y</font>] or  not  <a href="classes1.html#K6">the_rank_of</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>];<br/>
<a NAME="E30:1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E31:1_1"/><i><font color="Green">E138</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>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 ;<br/>

<a NAME="E2:1_1_5"/><i><font color="Green">E139</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E3:1_1_5"/>
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:1_1_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T74">CLASSES1:74</a></i>;<br/>
<a NAME="E5:1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then 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="E7:1_1_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E8:1_1_5"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> ]
 <b>and </b><br/><a NAME="E9:1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E10:1_1_5"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">y</font>] or  not  <a href="classes1.html#K6">the_rank_of</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E11:1_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E32:1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
