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

<span class="kw">let </span><font color="Maroon" title="c1">l</font> be   <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> holds <br/>1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Olive" title="b1">a</font> ) implies 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:136"/><span class="lab"><font color="Green" title="E75">Z0</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> holds <br/>1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">U</font> =  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>;<br/>
<a NAME="E2:136"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E3:136"/><span class="kw">then </span>
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D11" title="ORDINAL1:def.11">ORDINAL1:def 11</a></span>;<br/>
<a NAME="E4:136"/><span class="kw">then </span>
<font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E5:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A1</font></span>: 
<font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a></span>;<br/>
<a NAME="E6:136"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T57" target="_self" title="ORDINAL6:th.57">Unc1</a></span>;<br/>
<a NAME="E7:136"/>
 <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">g</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:136"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="ref" href="ordinal6.html#T62" target="_self" title="ORDINAL6:th.62">Th23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">o</font> =  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  as   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T57" target="_self" title="ORDINAL6:th.57">Unc1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> = <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c4">o</font>;<br/>
<a NAME="E10:136"/><span class="lab"><font color="Green" title="E78">A4</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal6.html#K3" title="ORDINAL6:func.3">criticals</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="ordinal6.html#K6" title="ORDINAL6:func.6">exp</a> <font color="Maroon" title="c4">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T74" target="_self" title="ORDINAL6:th.74">Th12</a></span>;<br/>
<a NAME="E11:136"/><span class="lab"><font color="Green" title="E79">B0</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">B1</font></span>: 
<font color="Maroon" title="c1">l</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="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E81">A5</font></span>: <font color="Maroon" title="c3">g</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c1">l</font> = 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:136"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="ordinal6.html#T39" target="_self" title="ORDINAL6:th.39">Th09</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</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">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">l</font></span>)</span></span>)</span>

;<br/>
<a NAME="E14:136"/>
<font color="Maroon" title="c1">l</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="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:136"><span class="lab"><font color="Green" title="E80">B1</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E15:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">B2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">l</font></span>)</span> implies <font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:136_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:136_2"/><span class="lab"><font color="Green" title="E83">A6</font></span>: 
( <font color="Maroon" title="c7">y</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="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">l</font></span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136_2"><span class="lab"><font color="Green" title="E83">A6</font></span></a></span>;<br/><a NAME="E5:136_2"/><span class="lab"><font color="Green" title="E84">A7</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136_2"><span class="lab"><font color="Green" title="E83">A6</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E6:136_2"/>
<font color="Maroon" title="c8">y</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="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:136"><span class="lab"><font color="Green" title="E80">B1</font></span></a>, <a class="txt" href="#E15:136"><span class="lab"><font color="Green" title="E82">B2</font></span></a>, <a class="txt" href="#E3:136_2"><span class="lab"><font color="Green" title="E83">A6</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><a NAME="E7:136_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> &amp;  <a href="ordinal4.html#K3" title="ORDINAL4:func.3">1-element_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">l</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:136"><span class="lab"><font color="Green" title="E79">B0</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/><a NAME="E8:136_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:136"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E5:136_2"><span class="lab"><font color="Green" title="E84">A7</font></span></a>, <a class="ref" href="ordinal6.html#T67" target="_self" title="ORDINAL6:th.67">Th26</a></span>;<br/><a NAME="E9:136_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E75">Z0</font></span></a>, <a class="txt" href="#E15:136"><span class="lab"><font color="Green" title="E82">B2</font></span></a>, <a class="txt" href="#E3:136_2"><span class="lab"><font color="Green" title="E83">A6</font></span></a></span>;<br/><a NAME="E10:136_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:136"><span class="lab"><font color="Green" title="E82">B2</font></span></a>, <a class="txt" href="#E3:136_2"><span class="lab"><font color="Green" title="E83">A6</font></span></a>, <a class="ref" href="ordinal6.html#T12" target="_self" title="ORDINAL6:th.12">LemE2</a></span>;<br/><span class="kw">hence </span><a NAME="E11:136_2"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:136"/>
1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:136"><span class="lab"><font color="Green" title="E81">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T76" title="ZFMISC_1:th.76">ZFMISC_1:76</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E18:136"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E20:136"/><span class="lab"><font color="Green" title="E83">A8</font></span>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c7">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T47" title="ORDINAL5:th.47">ORDINAL5:47</a></span>;<br/>
<a NAME="E21:136"/>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E75">Z0</font></span></a>, <a class="txt" href="#E20:136"><span class="lab"><font color="Green" title="E83">A8</font></span></a></span>;<br/>
<a NAME="E22:136"/><span class="kw">then </span>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:136"><span class="lab"><font color="Green" title="E83">A8</font></span></a>, <a class="ref" href="ordinal6.html#T72" target="_self" title="ORDINAL6:th.72">Th15a</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:136"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 <a href="ordinal6.html#K13" title="ORDINAL6:func.13">-Veblen</a> <font color="Maroon" title="c1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:136"><span class="lab"><font color="Green" title="E83">A8</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
