<?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">U</font> being   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st <font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; (  for <font color="Olive" title="b2">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="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">U</font> <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> <font color="Olive" title="b2">a</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Olive" title="b1">U</font> ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">U</font> <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> <font color="Maroon" title="c1">l</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Olive" title="b1">U</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">U</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</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="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; (  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/><span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Olive" title="b1">a</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> ) implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:115"/><span class="lab"><font color="Green" title="E59">Z0</font></span>: 
<font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <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 <br/>( <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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Olive" title="b1">a</font> is  not   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> ) or <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:115"/><span class="lab"><font color="Green" title="E60">Z1</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/><span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Olive" title="b1">a</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font></span><br/>

<a NAME="E3:115"/><span class="lab"><font color="Green" title="E61">A0</font></span>: 
<font color="Maroon" title="c1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E59">Z0</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<a NAME="E4:115"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal6.html#K7" title="ORDINAL6:func.7">criticals</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D15" target="_self" title="ORDINAL6:def.15">V</a></span>;<br/>
<a NAME="E5:115"/><span class="lab"><font color="Green" title="E63">A3</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="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#D15" target="_self" title="ORDINAL6:def.15">V</a></span>;<br/>
<a NAME="E6:115"/>
<font color="Maroon" title="c1">l</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115"><span class="lab"><font color="Green" title="E61">A0</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E7:115"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="txt" href="#E5:115"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E8:115"/><span class="lab"><font color="Green" title="E65">YY</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">G</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font>;<br/>
<a NAME="E1:115_1"/><span class="lab"><font color="Green" title="E66">H1</font></span>: 
 <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 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 <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E60">Z1</font></span></a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E3:115_1"/><span class="lab"><font color="Green" title="E67">H2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<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="E4:115_1"/><span class="lab"><font color="Green" title="E68">H2a</font></span>: 
<font color="Maroon" title="c5">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E5:115_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f0</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E66">H1</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:115_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E67">H2</font></span></a>, <a class="ref" href="ordinal6.html#T49" target="_self" title="ORDINAL6:th.49">ThA5</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="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">c</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">c</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c6">c</font> is   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> )</span><br/><span class="kw">assume </span><a NAME="E1:115_1_1"/><span class="lab"><font color="Green" title="E69">C1</font></span>: 
<font color="Maroon" title="c6">c</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c6">c</font> is   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font></span><br/><a NAME="E2:115_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c6">c</font> is   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:115_1_1"><span class="lab"><font color="Green" title="E69">C1</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E60">Z1</font></span></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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">lg</font> =  <a href="ordinal6.html#K8" title="ORDINAL6:func.8">lims</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> as   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E59">Z0</font></span></a>, <a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="ordinal6.html#T56" target="_self" title="ORDINAL6:th.56">ThL1</a></span>;<br/>
<a NAME="E9:115_1"/><span class="lab"><font color="Green" title="E69">H3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<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="E10:115_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">lg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">lg</font> or <font color="Maroon" title="c7">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:115_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">lg</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="c7">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:115_1_2"/><span class="lab"><font color="Green" title="E70">G1</font></span>: 
( <font color="Maroon" title="c8">x</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="c6">lg</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">lg</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c8">x</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="c9">x</font> = <font color="Maroon" title="c8">x</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="#E3:115_1_2"><span class="lab"><font color="Green" title="E70">G1</font></span></a></span>;<br/>
<a NAME="E5:115_1_2"/><span class="lab"><font color="Green" title="E71">G2</font></span>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:115_1"><span class="lab"><font color="Green" title="E69">H3</font></span></a>, <a class="txt" href="#E3:115_1_2"><span class="lab"><font color="Green" title="E70">G1</font></span></a>, <a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Y</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; (  for <font color="Olive" title="b2">f</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st <font color="Olive" title="b2">f</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Olive" title="b2">f</font> ) ) </span> } </span> ;<br/>
<a NAME="E6:115_1_2"/><span class="lab"><font color="Green" title="E72">G4</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; (  for <font color="Olive" title="b2">f</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st <font color="Olive" title="b2">f</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Olive" title="b2">f</font> ) ) </span> } </span>  is  <a href="ordinal6.html#V1" title="ORDINAL6:attr.1">ordinal-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T46" target="_self" title="ORDINAL6:th.46">ThA1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">f</font> be   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</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="c11">f</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> implies <font color="Maroon" title="c7">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c11">f</font> )</span><br/><span class="kw">assume </span><a NAME="E1:115_1_2_1"/>
<font color="Maroon" title="c11">f</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c7">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c11">f</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:115_1_2_1"/><span class="lab"><font color="Green" title="E73">G3</font></span>: 
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c12">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:115_1_2_1"/>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c12">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1_2_1"><span class="lab"><font color="Green" title="E73">G3</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:115_1_2_1"/>
<font color="Maroon" title="c7">y</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E59">Z0</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E60">Z1</font></span></a>, <a class="txt" href="#E3:115_1_2_1"><span class="lab"><font color="Green" title="E73">G3</font></span></a>, <a class="txt" href="#E3:115_1_2"><span class="lab"><font color="Green" title="E70">G1</font></span></a>, <a class="txt" href="#E5:115_1_2"><span class="lab"><font color="Green" title="E71">G2</font></span></a>, <a class="ref" href="ordinal6.html#T61" target="_self" title="ORDINAL6:th.61">Th53</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>
<a NAME="E8:115_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; (  for <font color="Olive" title="b2">f</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st <font color="Olive" title="b2">f</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Olive" title="b2">f</font> ) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E67">H2</font></span></a>, <a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E68">H2a</font></span></a>, <a class="txt" href="#E5:115_1_2"><span class="lab"><font color="Green" title="E71">G2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:115_1_2"/>
<font color="Maroon" title="c7">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E6:115_1_2"><span class="lab"><font color="Green" title="E72">G4</font></span></a>, <a class="ref" href="ordinal6.html#T19" target="_self" title="ORDINAL6:th.19">ThN1a</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>
<a NAME="E11:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">F1</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">lg</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T77" title="ZFMISC_1:th.77">ZFMISC_1:77</a></span>;<br/>
<a NAME="E12:115_1"/>
 <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">lg</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">lg</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:115_1_3"/><span class="lab"><font color="Green" title="E71">E1</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</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="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">lg</font></span><br/>

<a NAME="E2:115_1_3"/><span class="lab"><font color="Green" title="E72">E2</font></span>: 
<font color="Maroon" title="c7">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="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">X</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> </span> } </span> ;<br/>
<a NAME="E3:115_1_3"/>
 <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T51" title="CLASSES2:th.51">CLASSES2:51</a></span>;<br/>
<a NAME="E4:115_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">E3</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_3"><span class="lab"><font color="Green" title="E71">E1</font></span></a>, <a class="ref" href="ordinal1.html#T28" title="ORDINAL1:th.28">ORDINAL1:28</a></span>;<br/>
<a NAME="E5:115_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">E4</font></span>: 
<font color="Maroon" title="c6">lg</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:115_1"><span class="lab"><font color="Green" title="E69">H3</font></span></a>, <a class="ref" href="ordinal6.html#D12" target="_self" title="ORDINAL6:def.12">LIM</a></span>;<br/>
<a NAME="E6:115_1_3"/><span class="lab"><font color="Green" title="E75">E5</font></span>: 
<font color="Maroon" title="c5">f0</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E68">H2a</font></span></a>, <a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E66">H1</font></span></a></span>;<br/>
<a NAME="E7:115_1_3"/><span class="lab"><font color="Green" title="E76">E6</font></span>: 
<font color="Maroon" title="c5">f0</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f0</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_3"><span class="lab"><font color="Green" title="E72">E2</font></span></a>, <a class="txt" href="#E4:115_1_3"><span class="lab"><font color="Green" title="E73">E3</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E67">H2</font></span></a>, <a class="ref" href="ordinal2.html#D12" title="ORDINAL2:def.12">ORDINAL2:def 12</a></span>;<br/>
<a NAME="E8:115_1_3"/>
<font color="Maroon" title="c7">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">f0</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E67">H2</font></span></a>, <a class="txt" href="#E1:115_1_3"><span class="lab"><font color="Green" title="E71">E1</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/>
<a NAME="E9:115_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f0</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115_1_3"><span class="lab"><font color="Green" title="E76">E6</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<a NAME="E10:115_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">lg</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115_1_3"><span class="lab"><font color="Green" title="E74">E4</font></span></a>, <a class="txt" href="#E6:115_1_3"><span class="lab"><font color="Green" title="E75">E5</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:115_1_3"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">lg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:115_1"><span class="lab"><font color="Green" title="E69">H3</font></span></a>, <a class="txt" href="#E4:115_1_3"><span class="lab"><font color="Green" title="E73">E3</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</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>
<a NAME="E13:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">F2</font></span>: 
 <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:115_1"><span class="lab"><font color="Green" title="E70">F1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E14:115_1"/><span class="lab"><font color="Green" title="E72">F3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c7">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:115_1_4"/>
<font color="Maroon" title="c7">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:115_1_4"/><span class="lab"><font color="Green" title="E73">D1</font></span>: 
( <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <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="c8">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/>
<a NAME="E4:115_1_4"/>
<font color="Maroon" title="c7">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c5">f0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E66">H1</font></span></a>, <a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E68">H2a</font></span></a>, <a class="txt" href="#E3:115_1_4"><span class="lab"><font color="Green" title="E73">D1</font></span></a>, <a class="ref" href="ordinal6.html#T47" target="_self" title="ORDINAL6:th.47">ThA2</a></span>;<br/>
<a NAME="E5:115_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f0</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f0</font> <a href="ordinal6.html#K4" title="ORDINAL6:func.4">.</a> <font color="Maroon" title="c7">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:115_1_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 ; <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">assume </span><a NAME="E15:115_1"/>
 <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <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"> contradiction</span><br/>

<a NAME="E16:115_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<a NAME="E18:115_1"/><span class="lab"><font color="Green" title="E73">H5</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a></span>;<br/>
<a NAME="E19:115_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T12" title="CARD_1:th.12">CARD_1:12</a></span>;<br/>
<a NAME="E20:115_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:115_1"><span class="lab"><font color="Green" title="E73">H5</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">r</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">U</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:115_1"><span class="lab"><font color="Green" title="E72">F3</font></span></a>, <a class="ref" href="classes1.html#T1" title="CLASSES1:th.1">CLASSES1:1</a></span>;<br/>
<a NAME="E22:115_1"/>
 <a href="classes2.html#K3" title="CLASSES2:func.3">union</a> <font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 
;<br/>
<a NAME="E23:115_1"/><span class="kw">then </span>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></span>;<br/>
<a NAME="E24:115_1"/><span class="kw">then </span>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:115_1"><span class="lab"><font color="Green" title="E71">F2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:115_1"/>
contradiction
 ; <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>
<a NAME="E9:115"/><span class="lab"><font color="Green" title="E66">ZZ</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:115_2"/>
<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:115_2"/><span class="lab"><font color="Green" title="E67">C1</font></span>: 
( <font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <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="c4">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="c5">y</font> = <font color="Maroon" title="c4">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="#E3:115_2"><span class="lab"><font color="Green" title="E67">C1</font></span></a></span>;<br/>
<a NAME="E5:115_2"/><span class="lab"><font color="Green" title="E68">C2</font></span>: 
 <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="E6:115_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E3:115_2"><span class="lab"><font color="Green" title="E67">C1</font></span></a>, <a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="ordinal6.html#T47" target="_self" title="ORDINAL6:th.47">ThA2</a></span>;<br/>
<a NAME="E7:115_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></span>;<br/>
<a NAME="E8:115_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E60">Z1</font></span></a>, <a class="txt" href="#E5:115_2"><span class="lab"><font color="Green" title="E68">C2</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:115_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c2">U</font>
 <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>; <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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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> ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">l</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c3">a</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:115_3"/><span class="lab"><font color="Green" title="E67">B1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c3">a</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a> </span><br/><a NAME="E2:115_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="ordinal6.html#K11" title="ORDINAL6:func.11">-Veblen</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c3">a</font> is  <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E60">Z1</font></span></a>, <a class="txt" href="#E1:115_3"><span class="lab"><font color="Green" title="E67">B1</font></span></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>
<a NAME="E11:115"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E7:115"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="ordinal6.html#T53" target="_self" title="ORDINAL6:th.53">ThAA</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:115"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">U</font> <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> <font color="Maroon" title="c1">l</font> is   <a href="ordinal6.html#V2" title="ORDINAL6:attr.2">normal</a>  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c2">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:115"><span class="lab"><font color="Green" title="E65">YY</font></span></a>, <a class="txt" href="#E9:115"><span class="lab"><font color="Green" title="E66">ZZ</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</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>
