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

<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/><b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">Phi</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">D</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c4">L</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green" title="E14">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp; (  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> implies <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <b>and </b><br/><a NAME="E4:17"/><i><font color="Green" title="E15">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> holds <br/><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font></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">A</font>,<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font>)
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green" title="E16">A4</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  holds <br/><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S11" title="ORDINAL2:sch.11">ORDINAL2:sch 11</a>();<br/></i>
<a NAME="E6:17"/><i><font color="Green" title="E17">A5</font></i>: 
 <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35" title="ORDINAL4:th.35">ORDINAL4:35</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>;<br/>
<a NAME="E7:17"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="classes2.html#T62" title="CLASSES2:th.62">CLASSES2:62</a></i>;<br/>
<a NAME="E8:17"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>]
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E6:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E9:17"/><i><font color="Green" title="E20">A8</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E3:17_1"/>
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:17"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/>
<a NAME="E5:17_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="card_2.html#T13" title="CARD_2:th.13">CARD_2:13</a></i>;<br/>
<a NAME="E6:17_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="card_2.html#T2" title="CARD_2:th.2">CARD_2:2</a></i>;<br/>
<a NAME="E7:17_1"/><b>then </b><i><font color="Green" title="E22">A10</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E8:17_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">Phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:17_1"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_5.html#T48" title="FUNCT_5:th.48">FUNCT_5:48</a>, <a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/>
<a NAME="E10:17_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp;  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T9" title="ORDINAL2:th.9">ORDINAL2:9</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:17_1"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:17_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>
<a NAME="E13:17_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E7:17_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="zfrefle1.html#E11:17_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a>, <a class="ref" href="classes2.html#T63" title="CLASSES2:th.63">CLASSES2:63</a></i>;<br/>
<a NAME="E14:17_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="classes2.html#T66" title="CLASSES2:th.66">CLASSES2:66</a></i>;<br/>
<a NAME="E15:17_1"/><b>then </b>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:17_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="zf_refle.html#T28" title="ZF_REFLE:th.28">ZF_REFLE:28</a></i>;<br/>
<b>hence </b><a NAME="E16:17_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c5">a</font>]
 <b>by </b><i><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:17"/><i><font color="Green" title="E21">A12</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_2"/><i><font color="Green" title="E22">A13</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:17_2"/><i><font color="Green" title="E23">A14</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> holds <br/><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 ;<br/>

<a NAME="E3:17_2"/><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E4:17_2"/><b>then </b><i><font color="Green" title="E25">A16</font></i>: 
( <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="zfrefle1.html#E6:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="zfrefle1.html#E1:17_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a>, <a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<a NAME="E5:17_2"/><b>then </b><i><font color="Green" title="E26">A17</font></i>: 
( <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/>
<a NAME="E6:17_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1"/>
<font color="Maroon" title="c6">e</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_2_1"/><i><font color="Green" title="E27">A18</font></i>: 
( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">u</font> )
 <b>by </b><i><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="c8">u</font> = <font color="Maroon" title="c7">u</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:17_2_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:17_2_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:17_2"><i><font color="Green" title="E24">A15</font></i></a>, <a class="txt" href="zfrefle1.html#E4:17_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="zfrefle1.html#E3:17_2_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">u</font> = <font color="Maroon" title="c8">u</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E7:17_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:17_2_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:17_2_1"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E2:17_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="zfrefle1.html#E4:17_2"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="zfrefle1.html#E3:17_2_1"><i><font color="Green" title="E27">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:17_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_2"/><b>then </b>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E5:17_2"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="zfrefle1.html#T12" target="_self" title="ZFREFLE1:th.12">Th12</a>, <a class="ref" href="zf_refle.html#T28" title="ZF_REFLE:th.28">ZF_REFLE:28</a></i>;<br/>
<b>hence </b><a NAME="E8:17_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">a</font>]
 <b>by </b><i><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:17"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_3"/>
<font color="Maroon" title="c5">e</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">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_3"/><i><font color="Green" title="E22">A19</font></i>: 
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</font> )
 <b>by </b><i><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="c7">u</font> = <font color="Maroon" title="c6">u</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E3:17_3"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E5:17_3"/>
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S4" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>(<a class="txt" href="zfrefle1.html#E8:17"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="zfrefle1.html#E9:17"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="zfrefle1.html#E10:17"><i><font color="Green" title="E21">A12</font></i></a>);<br/></i>
<a NAME="E6:17_3"/><b>then </b>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
;<br/>
<b>hence </b><a NAME="E7:17_3"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:17_3"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">phi</font> = <font color="Maroon" title="c4">L</font> as    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="ordinal4.html#D3" title="ORDINAL4:def.3">ORDINAL4:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">phi</font>
;<br/>

<b>thus </b><a NAME="E13:17"/><i><font color="Green" title="E22">A20</font></i>: 
<font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon" title="c7">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:17_4"/><i><font color="Green" title="E23">A21</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">phi</font> )
 ;<br/>

<a NAME="E2:17_4"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c7">B</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E1:17_4"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</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>( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 implies <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E4:17_4"/><i><font color="Green" title="E24">A22</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>]
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35" title="ORDINAL4:th.35">ORDINAL4:35</a></i>;<br/>
<a NAME="E5:17_4"/><i><font color="Green" title="E25">A23</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">b</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_4_1"/><i><font color="Green" title="E26">A24</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b</font> implies <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font> )
 <b>and </b><br/><a NAME="E2:17_4_1"/><i><font color="Green" title="E27">A25</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c10">b</font>
 ;<br/>

<a NAME="E3:17_4_1"/>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E4:17_4_1"/><b>then </b>
( <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c10">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c10">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:17"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:17_4_1"/><b>then </b>
<font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c10">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E6:17_4_1"/>
<font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:17_4_1"><i><font color="Green" title="E26">A24</font></i></a>, <a class="txt" href="zfrefle1.html#E2:17_4_1"><i><font color="Green" title="E27">A25</font></i></a>, <a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:17_4"/><i><font color="Green" title="E26">A26</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">b</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">c</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">c</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">b</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_4_2"/><i><font color="Green" title="E27">A27</font></i>: 
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c10">b</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:17_4_2"/>
for <font color="Olive" title="b1">c</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">c</font>
 <b>and </b><br/><a NAME="E3:17_4_2"/><i><font color="Green" title="E28">A28</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b</font>
 ;<br/>

<a NAME="E4:17_4_2"/>
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E5:17_4_2"/><b>then </b>
( <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c10">b</font></span>)</span> &amp; <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c10">b</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="zfrefle1.html#E1:17_4_2"><i><font color="Green" title="E27">A27</font></i></a>, <a class="txt" href="zfrefle1.html#E3:17_4_2"><i><font color="Green" title="E28">A28</font></i></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></i>;<br/>
<b>hence </b><a NAME="E6:17_4_2"/>
<font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_4"/>
for <font color="Olive" title="b1">b</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S4" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>(<a class="txt" href="zfrefle1.html#E4:17_4"><i><font color="Green" title="E24">A22</font></i></a>, <a class="txt" href="zfrefle1.html#E5:17_4"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="zfrefle1.html#E6:17_4"><i><font color="Green" title="E26">A26</font></i></a>);<br/></i>
<a NAME="E8:17_4"/><b>then </b>
<font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:17_4"><i><font color="Green" title="E23">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:17_4"/>
<font color="Maroon" title="c5">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">B</font>
 ;<br/>


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

<b>thus </b><a NAME="E14:17"/>
<font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a><br/><b>let </b><font color="Maroon" title="c7">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:17_5"/><i><font color="Green" title="E23">A29</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">phi</font> &amp; <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">A</font> )
 ;<br/>

<a NAME="E2:17_5"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:17_5"/><b>then </b>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">A</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="zfrefle1.html#E13:17"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="zfrefle1.html#E1:17_5"><i><font color="Green" title="E23">A29</font></i></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E4:17_5"/>
<font color="Maroon" title="c7">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:17_5"><i><font color="Green" title="E23">A29</font></i></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></i>;<br/>


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

<b>thus </b><a NAME="E15:17"/>
<font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E6:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="zfrefle1.html#E7:17"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>

<b>thus </b><a NAME="E16:17"/>
for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<a NAME="E1:17_6"/>
 <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<b>hence </b><a NAME="E2:17_6"/>
<font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c6">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c3">Phi</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">D</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c6">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E4:17"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/>

<a NAME="E17:17"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<b>hence </b><a NAME="E18:17"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  implies <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c6">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="zfrefle1.html#E6:17"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>


</div>
