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

<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">U</font> being   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>  st  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">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></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"> (  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">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> )</span><br/>



<span class="kw">assume </span><a NAME="E1:116"/><span class="lab"><font color="Green" title="E60">Z0</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <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"> (  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">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> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> $1 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> );<br/>
<a NAME="E2:116"/><span class="lab"><font color="Green" title="E61">00</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E60">Z0</font></span></a>, <a class="txt" href="ordinal6.html#E78"><span class="lab"><font color="Green" title="E56">Th21</font></span></a></span>;<br/>
<a NAME="E3:116"/><span class="lab"><font color="Green" title="E62">01</font></span>: 
 for <font color="Olive" title="b1">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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">b</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">b</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">b</font>] )</span><br/>

<a NAME="E1:116_1"/>
<font color="Maroon" title="c3">b</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="c3">b</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/>
<a NAME="E2:116_1"/><span class="kw">then </span>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> implies <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:116_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">b</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">b</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E60">Z0</font></span></a>, <a class="txt" href="ordinal6.html#E79"><span class="lab"><font color="Green" title="E57">Th22</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="E4:116"/><span class="lab"><font color="Green" title="E63">02</font></span>: 
 for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b2">c</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">c</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">b</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">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b1">c</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">b</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:116_2"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E2:116_2"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 for <font color="Olive" title="b1">c</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>]
 <span class="kw">and </span><br/><a NAME="E3:116_2"/><span class="lab"><font color="Green" title="E66">A3</font></span>: 
<font color="Maroon" title="c3">b</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"> <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">b</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</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="c4">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> )</span><br/><span class="kw">assume </span><a NAME="E1:116_2_1"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">b</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="c4">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></span><br/><a NAME="E2:116_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_2"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:116_2_1"/>
<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="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116_2"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E1:116_2_1"><span class="lab"><font color="Green" title="E67">A4</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">hence </span><a NAME="E5:116_2"/>
<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">b</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="#E1:116_2"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E3:116_2"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="ordinal6.html#E80"><span class="lab"><font color="Green" title="E58">Th61</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="E5:116"/>
 for <font color="Olive" title="b1">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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E61">00</font></span></a>, <a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E62">01</font></span></a>, <a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E63">02</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:116"/>
(  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">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"> verum</span><br/>


</div>
