<?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">L</font> be   <a href="zf_refle.html#NM2" title="ZF_REFLE:NM.2">DOMAIN-Sequence</a> of <font color="Maroon" title="c1">W</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <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="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font>
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green" title="E22">A3</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="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="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</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">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> ex <font color="Olive" title="b2">phi</font> being   <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">phi</font> &amp; <font color="Olive" title="b2">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Olive" title="b2">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> &amp; ( for <font color="Olive" title="b3">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="b2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a</font> holds <br/>for <font color="Olive" title="b4">va</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b4">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> ) ) );<br/>
<a NAME="E4:28"/><i><font color="Green" title="E23">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="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> &amp; <font color="Maroon">S<sub>2</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="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c3">e</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">phi</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E4:28_1"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> &amp; ( 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" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">va</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E2:28"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E3:28"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="zf_refle.html#T29" title="ZF_REFLE:th.29">ZF_REFLE:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">phi</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c6">u</font>
;<br/>

<a NAME="E6:28_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">phi</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">phi</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> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#D3" title="ORDINAL4:def.3">ORDINAL4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:28_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c4">H</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">phi</font>
;<br/>

<b>thus </b><a NAME="E8:28_1"/>
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">phi</font> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> &amp; ( 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" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">va</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E4:28_1"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">Phi</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:28"/><i><font color="Green" title="E24">A6</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  &amp;  <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>and </b><br/><a NAME="E7:28"/><i><font color="Green" title="E25">A7</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="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">Phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="zfrefle1.html#E4:28"><i><font color="Green" title="E23">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">Phi</font> = <font color="Maroon" title="c3">Phi</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</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="txt" href="zfrefle1.html#E6:28"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E9:28"/>
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<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>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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#E1:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="classes2.html#T67" title="CLASSES2:th.67">CLASSES2:67</a></i>;<br/>
<a NAME="E10:28"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> &amp;  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<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> &amp;  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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="ref" href="classes2.html#T65" title="CLASSES2:th.65">CLASSES2:65</a>, <a class="ref" href="zf_lang1.html#T146" title="ZF_LANG1:th.146">ZF_LANG1:146</a></i>;<br/>
<a NAME="E11:28"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <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> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a></i>;<br/>
<a NAME="E12:28"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <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="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">phi</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E14:28"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>and </b><br/><a NAME="E15:28"/>
<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>and </b><br/><a NAME="E16:28"/><i><font color="Green" title="E27">A9</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> 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="c4">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"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> ,<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>
 <b>and </b><br/><a NAME="E17:28"/><i><font color="Green" title="E28">A10</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>  holds <br/><font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">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="Olive" title="b1">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfrefle1.html#T14" target="_self" title="ZFREFLE1:th.14">Th14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">phi</font>
;<br/>

<b>thus </b><a NAME="E18:28"/>
( <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c5">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E14:28"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>

<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/>

<b>assume </b><a NAME="E19:28"/><i><font color="Green" title="E29">A11</font></i>: 
( <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> <font color="Maroon" title="c6">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> )
 ;<br/>

<b>thus </b><a NAME="E20:28"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="zf_refle.html#T24" title="ZF_REFLE:th.24">ZF_REFLE:24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D3" target="_self" title="ZFREFLE1:def.3">ZFREFLE1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c7">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>let </b><font color="Maroon" title="c8">va</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>;<br/>



<b>set </b><font color="Maroon" title="c9">M</font> =  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>;<br/>
<a NAME="E21:28"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c7">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">H1</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>, <font color="Maroon" title="c11">xi</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E23:28"/><i><font color="Green" title="E31">A13</font></i>: 
( <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> &amp; <font color="Maroon" title="c4">Phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xi</font> &amp; <font color="Maroon" title="c11">xi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c11">xi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> &amp; ( 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" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">va</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c10">H1</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c10">H1</font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E7:28"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/>
<a NAME="E24:28"/><i><font color="Green" title="E32">A14</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="c11">xi</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</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>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c11">xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E25:28"/><i><font color="Green" title="E33">A15</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="E26:28"/><i><font color="Green" title="E34">A16</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>3</sub></font>[<font color="Olive" title="b1">a</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">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="E1:28_2"/>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">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="c11">xi</font> &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</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"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:28_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">H</font>,<span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">Phi</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">Phi</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">H</font>,<span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xi</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="c12">a</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">H</font>,<span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font></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"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> ,<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="c12">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> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E6:28"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="zfrefle1.html#E21:28"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="zfrefle1.html#E23:28"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:28_2"/><b>then </b>
<font color="Maroon" title="c11">xi</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="c12">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">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"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> ,<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="c12">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>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:28_2"/><b>then </b>
<font color="Maroon" title="c11">xi</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="c12">a</font></span>)</span> <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="c12">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="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="c4">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"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> ,<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="c12">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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:28_2"/><b>then </b>
( <font color="Maroon" title="c11">xi</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="c12">a</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"><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="c12">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="c4">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"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> ,<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="c12">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> &amp; <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="c12">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="c12">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="c4">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"><a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> ,<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="c12">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#E16:28"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E6:28_2"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">a</font>] implies <font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">a</font>] )
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:28"/><i><font color="Green" title="E35">A17</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>3</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="E1:28_3"/><i><font color="Green" title="E36">A18</font></i>: 
( <font color="Maroon" title="c12">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="c12">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:28_3"/><i><font color="Green" title="E37">A19</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="c12">a</font> &amp; <font color="Olive" title="b1">b</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>  holds <br/><font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">b</font>
 <b>and </b><br/><a NAME="E3:28_3"/><i><font color="Green" title="E38">A20</font></i>: 
<font color="Maroon" title="c12">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> 
 ;<br/>

<i><font color="Green" title="E39">A21</font></i>: <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> = 
 <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="c12">a</font></span>)</span>
<b>by </b><i><a class="txt" href="zfrefle1.html#E17:28"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="zfrefle1.html#E1:28_3"><i><font color="Green" title="E36">A18</font></i></a></i>
<br/>.= 
 <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="c5">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>

;<br/>
<b>let </b><font color="Maroon" title="c13">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="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a><br/>

<b>assume </b><a NAME="E5:28_3"/><i><font color="Green" title="E40">A22</font></i>: 
<font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 ;<br/>

<a NAME="E6:28_3"/>
<font color="Maroon" title="c12">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="c11">xi</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E7:28_3"/><b>then </b><i><font color="Green" title="E41">A23</font></i>: 
( <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c12">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="c11">xi</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E23:28"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="zfrefle1.html#E1:28_3"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="zfrefle1.html#E3:28_3"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></i>;<br/>
<a NAME="E8:28_3"/><b>then </b><i><font color="Green" title="E42">A24</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>then </b><font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> = 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span>
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:28_3"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="zfrefle1.html#E3:28_3"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="zfrefle1.html#E7:28_3"><i><font color="Green" title="E41">A23</font></i></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></i>
<br/>.= 
 <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="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon" title="c14">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E11:28_3"/><i><font color="Green" title="E43">A25</font></i>: 
( <font color="Maroon" title="c14">B</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="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span> &amp; <font color="Maroon" title="c13">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c14">B</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E5:28_3"><i><font color="Green" title="E40">A22</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="c15">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:28_3"/><i><font color="Green" title="E44">A26</font></i>: 
( <font color="Maroon" title="c15">e</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="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span> &amp; <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">e</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E11:28_3"><i><font color="Green" title="E43">A25</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="c16">e</font> = <font color="Maroon" title="c15">e</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E13:28_3"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E15:28_3"/>
<font color="Maroon" title="c12">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="E16:28_3"/><b>then </b>
<font color="Maroon" title="c16">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#E8:28_3"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="zfrefle1.html#E13:28_3"><i><font color="Green" title="E44">A26</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="c17">e</font> = <font color="Maroon" title="c16">e</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="E18:28_3"/>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</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="txt" href="zfrefle1.html#E1:28_3"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="zfrefle1.html#E8:28_3"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="zfrefle1.html#E13:28_3"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E19:28_3"/><b>then </b>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</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;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c17">e</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="c17">e</font> &amp; <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="c17">e</font></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> <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="c12">a</font></span>)</span> &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c17">e</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="c11">xi</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E20:28_3"/><b>then </b>
( <font color="Maroon" title="c11">xi</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="c17">e</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="c17">e</font></span>)</span> &amp; <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">xi</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="c17">e</font></span>)</span> &amp; <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="c17">e</font></span>)</span> <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="c12">a</font> &amp; <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c17">e</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E23:28"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="zfrefle1.html#E2:28_3"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="zfrefle1.html#E4:28_3"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="zfrefle1.html#E13:28_3"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a></i>;<br/>
<a NAME="E21:28_3"/><b>then </b>
( <font color="Maroon" title="c13">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">xi</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="c17">e</font></span>)</span> &amp; <font color="Maroon" title="c11">xi</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="c17">e</font></span>)</span> <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="c12">a</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E11:28_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E22:28_3"/>
<font color="Maroon" title="c13">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="c12">a</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:28"/>
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>3</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#E25:28"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="zfrefle1.html#E26:28"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="zfrefle1.html#E27:28"><i><font color="Green" title="E35">A17</font></i></a>);<br/></i>
<a NAME="E29:28"/><b>then </b>
( <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c11">xi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E19:28"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="zfrefle1.html#E23:28"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="zfrefle1.html#E24:28"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></i>;<br/>
<a NAME="E30:28"/><b>then </b>
<font color="Maroon" title="c11">xi</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> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E31:28"/>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">!</a> <font color="Maroon" title="c8">va</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E19:28"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="zfrefle1.html#E23:28"><i><font color="Green" title="E31">A13</font></i></a></i>;<br/>


</div>
