<?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">b</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/><b>let </b><font color="Maroon" title="c3">phi</font> be    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E22">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> &amp; <font color="Maroon" title="c3">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c3">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 ;<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="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>) <b>-&gt; </b>   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> =  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> $2</span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> =  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">nu</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="E3:35"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c4">nu</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> <font color="Maroon" title="c2">b</font>
 <b>and </b><br/><a NAME="E4:35"/><i><font color="Green" title="E24">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> holds  <font color="Maroon" title="c4">nu</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font>)
 <b>and </b><br/><a NAME="E5:35"/>
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="c4">nu</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="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">nu</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="zf_refle.html#S4" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon" title="c5">xi</font> = <font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">a</font> =  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span>;<br/>
<a NAME="E6:35"/><i><font color="Green" title="E25">A4</font></i>: 
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</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> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">nu</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> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a>, <a class="ref" href="ordinal4.html#D3" title="ORDINAL4:def.3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E7:35"/><b>then </b><i><font color="Green" title="E26">A5</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">nu</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:35"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E9:35"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</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> <font color="Maroon" title="c4">nu</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">nu</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="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a>, <a class="ref" href="ordinal4.html#D3" title="ORDINAL4:def.3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E10:35"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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;  <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="E11:35"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</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="E12:35"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="zfrefle1.html#T12" target="_self" title="ZFREFLE1:th.12">Th12</a></i>;<br/>
<a NAME="E13:35"/><b>then </b>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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="zf_refle.html#T28" title="ZF_REFLE:th.28">ZF_REFLE:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">a</font> =  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> 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="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">a</font>
;<br/>

<a NAME="E15:35"/>
(  <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>  &amp;  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">nu</font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35" title="ORDINAL4:th.35">ORDINAL4:35</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E16:35"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c2">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="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:35"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E1"><i><font color="Green" title="E1">Lm1</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="E17:35"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>

<a NAME="E18:35"/><i><font color="Green" title="E29">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:35_1"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c8">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">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E31">A10</font></i>: 
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> &amp; <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <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="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="c10">u</font> = <font color="Maroon" title="c9">u</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:35_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:35_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 
;<br/>
<b>hence </b><a NAME="E6:35_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:35_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:35"/><i><font color="Green" title="E30">A11</font></i>: 
<font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="c9">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:35_2"/><i><font color="Green" title="E31">A12</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> )
 ;<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="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> &amp; $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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1</span>)</span> );<br/>
<a NAME="E2:35_2"/><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E3:35_2"/><i><font color="Green" title="E33">A14</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_2_1"/><i><font color="Green" title="E34">A15</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> &amp; <font color="Maroon" title="c10">C</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>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:35_2_1"/><i><font color="Green" title="E35">A16</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</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/>

<a NAME="E3:35_2_1"/><i><font color="Green" title="E36">A17</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span> <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#E6:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green" title="E35">A16</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="c11">asc</font> = <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span> 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="E5:35_2_1"/>
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<a NAME="E6:35_2_1"/><b>then </b>
<font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">asc</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T49" title="ORDINAL2:th.49">ORDINAL2:49</a></i>;<br/>
<a NAME="E7:35_2_1"/><b>then </b><i><font color="Green" title="E37">A18</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</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="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_2_1"><i><font color="Green" title="E36">A17</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="c12">ac</font> = <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">C</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="E9:35_2_1"/>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">asc</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font> &amp; ( <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">C</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>  ) )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E10:35_2_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span> ) or <font color="Maroon" title="c10">C</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="txt" href="zfrefle1.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E4:35"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="zfrefle1.html#E1:35_2_1"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="zfrefle1.html#E7:35_2_1"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></i>;<br/>
<a NAME="E11:35_2_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">asc</font> ) or <font color="Maroon" title="c10">C</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="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E12:35_2_1"/><b>then </b><i><font color="Green" title="E38">A19</font></i>: 
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">asc</font> ) or <font color="Maroon" title="c10">C</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="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2_1_1"/>
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:35_2_1_1"/><b>then </b><i><font color="Green" title="E39">A20</font></i>: 
( <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c12">ac</font> &amp; <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/><a NAME="E3:35_2_1_1"/><b>then </b>
( <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span> &amp; <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">ac</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">asc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">asc</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E4:35"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="zfrefle1.html#E7:35_2_1"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></i>;<br/><b>hence </b><a NAME="E4:35_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E2:35_2_1_1"><i><font color="Green" title="E39">A20</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:35_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E12:35_2_1"><i><font color="Green" title="E38">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:35_2"/><i><font color="Green" title="E34">A21</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <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>  &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="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  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>2</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:35_2_2"/><i><font color="Green" title="E35">A22</font></i>: 
( <font color="Maroon" title="c10">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>  &amp; <font color="Maroon" title="c10">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E2:35_2_2"/><b>then </b>
 <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="c10">B</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<a NAME="E3:35_2_2"/><b>then </b>
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c10">B</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35_2_2"><i><font color="Green" title="E35">A22</font></i></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a>, <a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></i>;<br/>
<b>hence </b><a NAME="E4:35_2_2"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <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;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] ) or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">B</font>] )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:35_2"/><i><font color="Green" title="E35">A23</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="zfrefle1.html#E2:35_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="zfrefle1.html#E4:35_2"><i><font color="Green" title="E34">A21</font></i></a>);<br/></i>
<a NAME="E6:35_2"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</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>  )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="ordinal3.html#T31" title="ORDINAL3:th.31">ORDINAL3:31</a></i>;<br/>
<b>hence </b><a NAME="E7:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:35_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="zfrefle1.html#E5:35_2"><i><font color="Green" title="E35">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:35"/><i><font color="Green" title="E31">A24</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E21:35"/><b>then </b><i><font color="Green" title="E32">A25</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">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="E22:35"/><b>then </b><i><font color="Green" title="E33">A26</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c7">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="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="zfrefle1.html#E16:35"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="zfrefle1.html#E19:35"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a>, <a class="ref" href="ordinal4.html#T16" title="ORDINAL4:th.16">ORDINAL4:16</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="txt" href="zfrefle1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E23:35"/><b>then </b>
( <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <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="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">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"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></i>;<br/>
<a NAME="E24:35"/><b>then </b><i><font color="Green" title="E34">A27</font></i>: 
<font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">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="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a></i>;<br/>
<b>thus </b><a NAME="E25:35"/>
<font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="zf_refle.html#D1" title="ZF_REFLE:def.1">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E1:35_3"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:35_3"/><i><font color="Green" title="E35">A28</font></i>: 
( <font color="Maroon" title="c9">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="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> &amp; <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E24:35"><i><font color="Green" title="E34">A27</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="c10">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:35_3"/><i><font color="Green" title="E36">A29</font></i>: 
( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E22:35"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_3"><i><font color="Green" title="E35">A28</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="c11">e</font> = <font color="Maroon" title="c10">e</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E5:35_3"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:35_3"/><i><font color="Green" title="E37">A30</font></i>: 
( <font color="Maroon" title="c12">C</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">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> &amp; <font color="Maroon" title="c11">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">C</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E5:35_3"><i><font color="Green" title="E36">A29</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="c13">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:35_3"/><i><font color="Green" title="E38">A31</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">u</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35_3"><i><font color="Green" title="E37">A30</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="c14">u</font> = <font color="Maroon" title="c13">u</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E12:35_3"/>
<font color="Maroon" title="c14">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:35_3"/><b>then </b>
<font color="Maroon" title="c14">u</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#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">u</font> = <font color="Maroon" title="c14">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="ordinal4.html#D2" title="ORDINAL4:def.2">ORDINAL4:def 2</a></i>;<br/>
<a NAME="E15:35_3"/><i><font color="Green" title="E39">A32</font></i>: 
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E8:35_3"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E16:35_3"/>
( <font color="Maroon" title="c11">e</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c12">C</font> iff ( <font color="Maroon" title="c11">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c11">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">C</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E17:35_3"/><b>then </b>
( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">C</font> or ( <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font> ) )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E21:35"><i><font color="Green" title="E32">A25</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35_3"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="zfrefle1.html#E15:35_3"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<a NAME="E18:35_3"/><b>then </b><i><font color="Green" title="E40">A33</font></i>: 
( <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">C</font> or <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">C</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a></i>;<br/>
<a NAME="E19:35_3"/>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">nu</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="zfrefle1.html#E2"><i><font color="Green" title="E2">Lm2</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="E20:35_3"/><b>then </b><i><font color="Green" title="E41">A34</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <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="c15">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nu</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="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font> &amp; <font color="Maroon" title="c4">nu</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="c15">u</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="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> &amp; <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="zfrefle1.html#E22:35"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="zfrefle1.html#E5:35_3"><i><font color="Green" title="E36">A29</font></i></a>, <a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></i>;<br/>
<a NAME="E21:35_3"/><b>then </b>
( <font color="Maroon" title="c4">nu</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="c15">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:35"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="zfrefle1.html#E18:35_3"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E22:35_3"/><b>then </b>
( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">nu</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="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c4">nu</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="c15">u</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E20:35_3"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E23:35_3"/><b>then </b>
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E24:35_3"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:35_3"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


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

<b>thus </b><a NAME="E26:35"/>
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="zfrefle1.html#E20:35"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D5" target="_self" title="ZFREFLE1:def.5">ZFREFLE1:def 5</a><br/>

<b>thus </b><a NAME="E27:35"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c4">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c7">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">nu</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E7:35"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="zfrefle1.html#E18:35"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="zfrefle1.html#E19:35"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>


</div>
