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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; ( for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> $2</span>)</span> holds <br/> ex <font color="Olive" title="b4">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">n</font>,<font color="Olive" title="b2">X</font>,<font color="Olive" title="b4">Y</font>] ) ) &amp; ( for <font color="Olive" title="b2">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b3">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> $2</span>)</span> holds <br/> ex <font color="Olive" title="b5">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b2">O</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b4">n</font>,<font color="Olive" title="b3">X</font>,<font color="Olive" title="b5">Y</font>] ) ) holds <br/><font color="Olive" title="b1">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">O</font> ) );<br/>
<a NAME="E1:1_1"/><i><font color="Green" title="E2">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( $2 is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> $2</span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,$1,<font color="Olive" title="b2">y</font>] );<br/>
<a NAME="E1:1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
for <font color="Olive" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x'</font>,<font color="Olive" title="b2">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" title="c3">x'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( $2 is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; <font color="Olive" title="b1">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> $2</span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Maroon" title="c3">x'</font>,<font color="Olive" title="b1">y</font>] );<br/>
<a NAME="E2:1_1_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">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" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">i'</font> = <font color="Maroon" title="c4">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>thus </b><a NAME="E3:1_1_1_1_1"/>
 ex <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">o</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b1">o</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c3">x'</font>,<font color="Olive" title="b2">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" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:1_1_1_1_1_1"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">i'</font>,<font color="Maroon" title="c3">x'</font>,<font color="Maroon" title="c6">y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">o</font> =  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c6">y</font>;<br/>

<b>take </b>
<font color="Maroon" title="c6">y</font>
;<br/>

<b>thus </b><a NAME="E3:1_1_1_1_1_1"/>
<font color="Maroon" title="c7">o</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 ;<br/>

<a NAME="E4:1_1_1_1_1_1"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T81" title="CLASSES1:th.81">CLASSES1:81</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c7">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T73" title="CLASSES1:th.73">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" title="c4">i</font>,<font color="Maroon" title="c3">x'</font>,<font color="Maroon" title="c6">y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E2:1_1_1_1_1_1"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:1_1_1_1"/><i><font color="Green" title="E5">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1_1_1"/><i><font color="Green" title="E6">A7</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1" title="ZFREFLE1:sch.1">ZFREFLE1:sch 1</a>(<a class="txt" href="lattice5.html#E2:1_1_1_1"><i><font color="Green" title="E4">A4</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c5">o</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font></span>)</span>;<br/>

<b>thus </b><a NAME="E6:1_1_1_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c5">o</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c5">o</font></span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c3">x'</font>,<font color="Olive" title="b2">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" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:1_1_1_1_2"/><i><font color="Green" title="E7">A8</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>and </b><br/><a NAME="E3:1_1_1_1_2"/><i><font color="Green" title="E8">A9</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_1_2"/><i><font color="Green" title="E9">A10</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">x'</font>,<font color="Maroon" title="c7">y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E5:1_1_1_1"><i><font color="Green" title="E6">A7</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">y</font>
;<br/>

<b>thus </b><a NAME="E5:1_1_1_1_2"/>
<font color="Maroon" title="c5">o</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 ;<br/>

<a NAME="E6:1_1_1_1_2"/>
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:1_1_1_1"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:1_1_1_1_2"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E8:1_1_1_1_2"/><b>then </b><i><font color="Green" title="E10">A11</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">O</font> = <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice5.html#E2:1_1_1_1_2"><i><font color="Green" title="E7">A8</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_1_2"/><i><font color="Green" title="E11">A12</font></i>: 
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c8">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:1_1_1_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="classes1.html#T43" title="CLASSES1:th.43">CLASSES1:43</a></i>;<br/>
<a NAME="E11:1_1_1_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c8">O</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:1_1_1_1_2"><i><font color="Green" title="E8">A9</font></i></a>, <a class="ref" href="classes1.html#T81" title="CLASSES1:th.81">CLASSES1:81</a></i>;<br/>
<a NAME="E12:1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c5">o</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E10:1_1_1_1_2"><i><font color="Green" title="E11">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:1_1_1_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c5">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T81" title="CLASSES1:th.81">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" title="c6">m</font>,<font color="Maroon" title="c3">x'</font>,<font color="Maroon" title="c7">y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E4:1_1_1_1_2"><i><font color="Green" title="E9">A10</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:1_1_1"/><i><font color="Green" title="E4">A13</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1"/><i><font color="Green" title="E5">A14</font></i>: 
for <font color="Olive" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1" title="ZFREFLE1:sch.1">ZFREFLE1:sch 1</a>(<a class="txt" href="lattice5.html#E1:1_1_1"><i><font color="Green" title="E3">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> $1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">Y</font>] );<br/>
<a NAME="E5:1_1_1"/><i><font color="Green" title="E6">A15</font></i>: 
 ex <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">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" title="c4">O2</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:1_1_1_2"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">m</font>,<font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">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" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:1_1_1_2_1"/><i><font color="Green" title="E7">A16</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_2_1"/><i><font color="Green" title="E8">A17</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>and </b><br/><a NAME="E4:1_1_1_2_1"/><i><font color="Green" title="E9">A18</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:1_1_1_2_1"/><i><font color="Green" title="E10">A19</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c7">Y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E4:1_1_1"><i><font color="Green" title="E5">A14</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">Y</font>
;<br/>

<a NAME="E6:1_1_1_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:1_1_1"><i><font color="Green" title="E4">A13</font></i></a>, <a class="txt" href="lattice5.html#E1:1_1_1_2_1"><i><font color="Green" title="E7">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:1_1_1_2_1"><i><font color="Green" title="E8">A17</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E8:1_1_1_2_1"/><b>then </b><i><font color="Green" title="E11">A20</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">O2</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">O</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice5.html#E3:1_1_1_2_1"><i><font color="Green" title="E8">A17</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_2_1"/><i><font color="Green" title="E12">A21</font></i>: 
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c8">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">O2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:1_1_1_2_1"><i><font color="Green" title="E11">A20</font></i></a>, <a class="ref" href="classes1.html#T43" title="CLASSES1:th.43">CLASSES1:43</a></i>;<br/>
<a NAME="E11:1_1_1_2_1"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c8">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">O</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T81" title="CLASSES1:th.81">CLASSES1:81</a></i>;<br/>
<b>hence </b><a NAME="E12:1_1_1_2_1"/>
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">O2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:1_1_1_2_1"><i><font color="Green" title="E9">A18</font></i></a>, <a class="txt" href="lattice5.html#E10:1_1_1_2_1"><i><font color="Green" title="E12">A21</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="c6">m</font>,<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c7">Y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E5:1_1_1_2_1"><i><font color="Green" title="E10">A19</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:1_1_1"/><i><font color="Green" title="E7">A22</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">O2</font>] &amp; ( for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">O</font>] holds <br/><font color="Maroon" title="c4">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="lattice5.html#E5:1_1_1"><i><font color="Green" title="E6">A15</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c4">O2</font>
;<br/>

<b>thus </b><a NAME="E8:1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">O2</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E7:1_1_1"><i><font color="Green" title="E7">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1_1"/><i><font color="Green" title="E3">A23</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">y2</font>] holds <br/><font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_2"/><i><font color="Green" title="E4">A24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y1</font>]
 <b>and </b><br/><a NAME="E2:1_1_2"/><i><font color="Green" title="E5">A25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y2</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">O1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:1_1_2"/><i><font color="Green" title="E6">A26</font></i>: 
<font color="Maroon" title="c5">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font>
 <b>and </b><br/><a NAME="E5:1_1_2"/><i><font color="Green" title="E7">A27</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c5">O1</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">Y</font>] )
 <b>and </b><br/><a NAME="E6:1_1_2"/><i><font color="Green" title="E8">A28</font></i>: 
for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span> holds <br/> ex <font color="Olive" title="b4">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">O</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">n</font>,<font color="Olive" title="b2">X</font>,<font color="Olive" title="b4">Y</font>] ) ) holds <br/><font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:1_1_2"><i><font color="Green" title="E4">A24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:1_1_2"/><i><font color="Green" title="E9">A29</font></i>: 
<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font>
 <b>and </b><br/><a NAME="E9:1_1_2"/><i><font color="Green" title="E10">A30</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">Y</font>] )
 <b>and </b><br/><a NAME="E10:1_1_2"/><i><font color="Green" title="E11">A31</font></i>: 
for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">x</font></span>)</span> holds <br/> ex <font color="Olive" title="b4">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">O</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">n</font>,<font color="Olive" title="b2">X</font>,<font color="Olive" title="b4">Y</font>] ) ) holds <br/><font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:1_1_2"><i><font color="Green" title="E5">A25</font></i></a></i>;<br/>
<a NAME="E11:1_1_2"/><i><font color="Green" title="E12">A32</font></i>: 
<font color="Maroon" title="c5">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">O2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:1_1_2"><i><font color="Green" title="E8">A28</font></i></a>, <a class="txt" href="lattice5.html#E9:1_1_2"><i><font color="Green" title="E10">A30</font></i></a></i>;<br/>
<a NAME="E12:1_1_2"/>
<font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">O1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:1_1_2"><i><font color="Green" title="E7">A27</font></i></a>, <a class="txt" href="lattice5.html#E10:1_1_2"><i><font color="Green" title="E11">A31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:1_1_2"/>
<font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:1_1_2"><i><font color="Green" title="E6">A26</font></i></a>, <a class="txt" href="lattice5.html#E8:1_1_2"><i><font color="Green" title="E9">A29</font></i></a>, <a class="txt" href="lattice5.html#E11:1_1_2"><i><font color="Green" title="E12">A32</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:1_1"/><i><font color="Green" title="E4">A33</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1"/><i><font color="Green" title="E5">A34</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">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" title="E6">A35</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S1" title="RECDEF_1:sch.1">RECDEF_1:sch 1</a>(<a class="txt" href="lattice5.html#E1:1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="lattice5.html#E2:1_1"><i><font color="Green" title="E3">A23</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c2">beta</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ,$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,$2 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ] &amp; $2 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; ( for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ,$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Olive" title="b2">y</font>] or  not  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default">$2 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> ) ) );<br/>
<a NAME="E7:1_1"/><i><font color="Green" title="E7">A36</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:1_1_3"/><i><font color="Green" title="E8">A37</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_3"/><i><font color="Green" title="E9">A38</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1_3"/><i><font color="Green" title="E10">A39</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lattice5.html#E4:1_1_3"><i><font color="Green" title="E9">A38</font></i></a></i>;<br/>
<a NAME="E7:1_1_3"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:1_1_3"><i><font color="Green" title="E8">A37</font></i></a>, <a class="ref" href="classes1.html#T74" title="CLASSES1:th.74">CLASSES1:74</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">alfa</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:1_1_3"/><i><font color="Green" title="E11">A40</font></i>: 
<font color="Maroon" title="c7">alfa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font>
 <b>and </b><br/><a NAME="E10:1_1_3"/><i><font color="Green" title="E12">A41</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">alfa</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<a NAME="E11:1_1_3"/><i><font color="Green" title="E13">A42</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c7">alfa</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E10:1_1_3"><i><font color="Green" title="E12">A41</font></i></a>, <a class="ref" href="classes1.html#T73" title="CLASSES1:th.73">CLASSES1:73</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:1_1_3"/><i><font color="Green" title="E14">A43</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">g</font>
 <b>and </b><br/><a NAME="E14:1_1_3"/><i><font color="Green" title="E15">A44</font></i>: 
<font color="Maroon" title="c7">alfa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E9:1_1_3"><i><font color="Green" title="E11">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lattice5.html#E4:1_1"><i><font color="Green" title="E4">A33</font></i></a>, <a class="txt" href="lattice5.html#E13:1_1_3"><i><font color="Green" title="E14">A43</font></i></a></i>;<br/>
<a NAME="E16:1_1_3"/><i><font color="Green" title="E16">A45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">alfa</font>,<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:1_1"><i><font color="Green" title="E6">A35</font></i></a>, <a class="txt" href="lattice5.html#E14:1_1_3"><i><font color="Green" title="E15">A44</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">O2</font> = <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E18:1_1_3"/><i><font color="Green" title="E17">A46</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:1_1_3"><i><font color="Green" title="E10">A39</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> $1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ,<font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Olive" title="b1">y</font>] );<br/>
<a NAME="E19:1_1_3"/>
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c7">alfa</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:1_1_3"><i><font color="Green" title="E13">A42</font></i></a>, <a class="ref" href="classes1.html#T81" title="CLASSES1:th.81">CLASSES1:81</a></i>;<br/>
<a NAME="E20:1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c10">O2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ,<font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Olive" title="b1">y</font>] )
 
<b>by </b><i><a class="txt" href="lattice5.html#E16:1_1_3"><i><font color="Green" title="E16">A45</font></i></a>, <a class="txt" href="lattice5.html#E18:1_1_3"><i><font color="Green" title="E17">A46</font></i></a></i>;<br/>
<a NAME="E21:1_1_3"/><b>then </b><i><font color="Green" title="E18">A47</font></i>: 
 ex <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">O</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c11">O</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E23:1_1_3"/><i><font color="Green" title="E19">A48</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">O</font>]
 <b>and </b><br/><a NAME="E24:1_1_3"/><i><font color="Green" title="E20">A49</font></i>: 
for <font color="Olive" title="b1">O'</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">O'</font>] holds <br/><font color="Maroon" title="c11">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O'</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="lattice5.html#E21:1_1_3"><i><font color="Green" title="E18">A47</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c12">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E26:1_1_3"/><i><font color="Green" title="E21">A50</font></i>: 
<font color="Maroon" title="c12">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c11">O</font>
 <b>and </b><br/><a NAME="E27:1_1_3"/><i><font color="Green" title="E22">A51</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c12">Y</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E18:1_1_3"><i><font color="Green" title="E17">A46</font></i></a>, <a class="txt" href="lattice5.html#E23:1_1_3"><i><font color="Green" title="E19">A48</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c13">u</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">Y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>

<b>take </b><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c6">b</font>;<br/>

<b>thus </b><a NAME="E28:1_1_3"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E5:1_1_3"><i><font color="Green" title="E10">A39</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">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" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ,<font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c13">u</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ]
 <b>by </b><i><a class="txt" href="lattice5.html#E18:1_1_3"><i><font color="Green" title="E17">A46</font></i></a>, <a class="txt" href="lattice5.html#E27:1_1_3"><i><font color="Green" title="E22">A51</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>

<b>thus </b><a NAME="E30:1_1_3"/>
<font color="Maroon" title="c13">u</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>

<b>given </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E31:1_1_3"/><i><font color="Green" title="E23">A52</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ,<font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c15">y</font>]
 <b>and </b><br/><a NAME="E32:1_1_3"/><i><font color="Green" title="E24">A53</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
 ;<br/>

<a NAME="E33:1_1_3"/><i><font color="Green" title="E25">A54</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c12">Y</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E32:1_1_3"><i><font color="Green" title="E24">A53</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E34:1_1_3"/><i><font color="Green" title="E26">A55</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c12">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">O</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E26:1_1_3"><i><font color="Green" title="E21">A50</font></i></a>, <a class="ref" href="classes1.html#T73" title="CLASSES1:th.73">CLASSES1:73</a></i>;<br/>
<a NAME="E35:1_1_3"/>
<font color="Maroon" title="c15">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c15">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a></i>;<br/>
<a NAME="E36:1_1_3"/><b>then </b>
<font color="Maroon" title="c11">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c15">y</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E24:1_1_3"><i><font color="Green" title="E20">A49</font></i></a>, <a class="txt" href="lattice5.html#E31:1_1_3"><i><font color="Green" title="E23">A52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E37:1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice5.html#E33:1_1_3"><i><font color="Green" title="E25">A54</font></i></a>, <a class="txt" href="lattice5.html#E34:1_1_3"><i><font color="Green" title="E26">A55</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E10:1_1"/><i><font color="Green" title="E8">A56</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1" title="ZFREFLE1:sch.1">ZFREFLE1:sch 1</a>(<a class="txt" href="lattice5.html#E7:1_1"><i><font color="Green" title="E7">A36</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$2,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:1_1"/><i><font color="Green" title="E9">A57</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E13:1_1"/><i><font color="Green" title="E10">A58</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E14:1_1"/><i><font color="Green" title="E11">A59</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3" title="RECDEF_1:sch.3">RECDEF_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c4">f</font>
;<br/>

<b>thus </b><a NAME="E15:1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E12:1_1"><i><font color="Green" title="E9">A57</font></i></a></i>;<br/>

<b>thus </b><a NAME="E16:1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="lattice5.html#E13:1_1"><i><font color="Green" title="E10">A58</font></i></a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>;<br/>
<a NAME="E17:1_1"/>
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:1_1"><i><font color="Green" title="E4">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:1_1"/><b>then </b><i><font color="Green" title="E12">A60</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:1_1"><i><font color="Green" title="E5">A34</font></i></a>, <a class="txt" href="lattice5.html#E13:1_1"><i><font color="Green" title="E10">A58</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E19:1_1"/><i><font color="Green" title="E13">A61</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_4"/><i><font color="Green" title="E14">A62</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>
 ;<br/>

<a NAME="E2:1_1_4"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T74" title="CLASSES1:th.74">CLASSES1:74</a></i>;<br/>
<a NAME="E3:1_1_4"/><b>then </b><i><font color="Green" title="E15">A63</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:1_1_4"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <b>and </b><br/><a NAME="E7:1_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ]
 <b>and </b><br/><a NAME="E8:1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>and </b><br/><a NAME="E9:1_1_4"/><i><font color="Green" title="E16">A64</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>,<font color="Olive" title="b1">y</font>] or  not  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E10:1_1"><i><font color="Green" title="E8">A56</font></i></a>, <a class="txt" href="lattice5.html#E3:1_1_4"><i><font color="Green" title="E15">A63</font></i></a></i>;<br/>
<a NAME="E10:1_1_4"/><i><font color="Green" title="E17">A65</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:1_1"><i><font color="Green" title="E11">A59</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">o1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E12:1_1_4"/><i><font color="Green" title="E18">A66</font></i>: 
<font color="Maroon" title="c7">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font>
 <b>and </b><br/><a NAME="E13:1_1_4"/><i><font color="Green" title="E19">A67</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">o1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:1_1_4"><i><font color="Green" title="E14">A62</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:1_1_4"/><i><font color="Green" title="E20">A68</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">g</font>
 <b>and </b><br/><a NAME="E16:1_1_4"/><i><font color="Green" title="E21">A69</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E12:1_1_4"><i><font color="Green" title="E18">A66</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lattice5.html#E4:1_1"><i><font color="Green" title="E4">A33</font></i></a>, <a class="txt" href="lattice5.html#E15:1_1_4"><i><font color="Green" title="E20">A68</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">o2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E19:1_1_4"/><i><font color="Green" title="E22">A70</font></i>: 
<font color="Maroon" title="c10">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E20:1_1_4"/><i><font color="Green" title="E23">A71</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b3">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c10">o2</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">Y</font>] )
 <b>and </b><br/><a NAME="E21:1_1_4"/>
for <font color="Olive" title="b1">o</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b4">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">o</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">n</font>,<font color="Olive" title="b2">X</font>,<font color="Olive" title="b4">Y</font>] ) ) holds <br/><font color="Maroon" title="c10">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">o</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E6:1_1"><i><font color="Green" title="E6">A35</font></i></a></i>;<br/>
<a NAME="E22:1_1_4"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E13:1_1_4"><i><font color="Green" title="E19">A67</font></i></a>, <a class="txt" href="lattice5.html#E16:1_1_4"><i><font color="Green" title="E21">A69</font></i></a>, <a class="ref" href="classes1.html#T81" title="CLASSES1:th.81">CLASSES1:81</a></i>;<br/>
<a NAME="E23:1_1_4"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T73" title="CLASSES1:th.73">CLASSES1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E25:1_1_4"/><i><font color="Green" title="E24">A72</font></i>: 
<font color="Maroon" title="c11">YY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c10">o2</font>
 <b>and </b><br/><a NAME="E26:1_1_4"/><i><font color="Green" title="E25">A73</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c11">YY</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E20:1_1_4"><i><font color="Green" title="E23">A71</font></i></a></i>;<br/>
<a NAME="E27:1_1_4"/>
(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c11">YY</font>] or  not  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c11">YY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E9:1_1_4"><i><font color="Green" title="E16">A64</font></i></a></i>;<br/>
<a NAME="E28:1_1_4"/><b>then </b><i><font color="Green" title="E26">A74</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c5">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c11">YY</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E26:1_1_4"><i><font color="Green" title="E25">A73</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E29:1_1_4"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c11">YY</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">o2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E25:1_1_4"><i><font color="Green" title="E24">A72</font></i></a>, <a class="ref" href="classes1.html#T73" title="CLASSES1:th.73">CLASSES1:73</a></i>;<br/>
<a NAME="E30:1_1_4"/><b>then </b><i><font color="Green" title="E27">A75</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">o2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E10:1_1_4"><i><font color="Green" title="E17">A65</font></i></a>, <a class="txt" href="lattice5.html#E28:1_1_4"><i><font color="Green" title="E26">A74</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:1_1_4"/>
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:1_1"><i><font color="Green" title="E4">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E32:1_1_4"/><b>then </b>
<font color="Maroon" title="c10">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E19:1_1_4"><i><font color="Green" title="E22">A70</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E33:1_1_4"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E30:1_1_4"><i><font color="Green" title="E27">A75</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:1_1"/><i><font color="Green" title="E14">A76</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="lattice5.html#E18:1_1"><i><font color="Green" title="E12">A60</font></i></a>, <a class="txt" href="lattice5.html#E19:1_1"><i><font color="Green" title="E13">A61</font></i></a>);<br/></i>
<a NAME="E21:1_1"/><i><font color="Green" title="E15">A77</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E22:1_1"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E13:1_1"><i><font color="Green" title="E10">A58</font></i></a>, <a class="txt" href="lattice5.html#E20:1_1"><i><font color="Green" title="E14">A76</font></i></a></i>;<br/>
<a NAME="E23:1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T74" title="CLASSES1:th.74">CLASSES1:74</a></i>;<br/>
<a NAME="E24:1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E26:1_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>and </b><br/><a NAME="E27:1_1"/><i><font color="Green" title="E16">A78</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" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ]
 <b>and </b><br/><a NAME="E28:1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>and </b><br/><a NAME="E29:1_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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" title="b1">y</font>] or  not  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E10:1_1"><i><font color="Green" title="E8">A56</font></i></a>, <a class="txt" href="lattice5.html#E21:1_1"><i><font color="Green" title="E15">A77</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[$1,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>];<br/>
<a NAME="E30:1_1"/><i><font color="Green" title="E17">A79</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i><a class="txt" href="lattice5.html#E13:1_1"><i><font color="Green" title="E10">A58</font></i></a>, <a class="txt" href="lattice5.html#E14:1_1"><i><font color="Green" title="E11">A59</font></i></a>, <a class="txt" href="lattice5.html#E27:1_1"><i><font color="Green" title="E16">A78</font></i></a></i>;<br/>
<a NAME="E31:1_1"/><i><font color="Green" title="E18">A80</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 ;<br/>

<a NAME="E2:1_1_5"/><i><font color="Green" title="E19">A81</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E3:1_1_5"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E20:1_1"><i><font color="Green" title="E14">A76</font></i></a></i>;<br/>
<a NAME="E4:1_1_5"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T74" title="CLASSES1:th.74">CLASSES1:74</a></i>;<br/>
<a NAME="E5:1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:1_1_5"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>and </b><br/><a NAME="E8:1_1_5"/><i><font color="Green" title="E20">A82</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ]
 <b>and </b><br/><a NAME="E9:1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>and </b><br/><a NAME="E10:1_1_5"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b1">y</font>] or  not  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E10:1_1"><i><font color="Green" title="E8">A56</font></i></a>, <a class="txt" href="lattice5.html#E2:1_1_5"><i><font color="Green" title="E19">A81</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:1_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="lattice5.html#E14:1_1"><i><font color="Green" title="E11">A59</font></i></a>, <a class="txt" href="lattice5.html#E8:1_1_5"><i><font color="Green" title="E20">A82</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E32:1_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="lattice5.html#E30:1_1"><i><font color="Green" title="E17">A79</font></i></a>, <a class="txt" href="lattice5.html#E31:1_1"><i><font color="Green" title="E18">A80</font></i></a>);</i><br/>


</div>
