<?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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> is  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  implies  ex <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" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Olive" title="b1">b</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <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 $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Olive" title="b1">b</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>for <font color="Olive" title="b1">e</font> being  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> $1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>];<br/>
<a NAME="E1:71"/><span class="lab"><font color="Green" title="E48">A0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">e</font> be   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">e</font>] )</span><br/>

<a NAME="E1:71_1"/>
 <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="c2">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/>
<a NAME="E2:71_1"/><span class="kw">then </span>
 <a href="ordinal5.html#K2" title="ORDINAL5:func.2">first_epsilon_greater_than</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#D6" target="_self" title="ORDINAL5:def.6">FEGT</a></span>;<br/>
<a NAME="E3:71_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T36" target="_self" title="ORDINAL5:th.36">Th20</a>, <a class="ref" href="ordinal5.html#T41" target="_self" title="ORDINAL5:th.41">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:71_1"/>
( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">e</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:71"/><span class="lab"><font color="Green" title="E49">A1</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="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">c</font>] implies <font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">c</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_2"/><span class="lab"><font color="Green" title="E50">Z0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">c</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">c</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">c</font></span>)</span> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E2:71_2"/><span class="lab"><font color="Green" title="E51">Z1</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]</span><br/>

<span class="lab"><font color="Green" title="E52">Z2</font></span>:  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T42" target="_self" title="ORDINAL5:th.42">Th31</a></span>
<br/>.= 
 <a href="ordinal5.html#K2" title="ORDINAL5:func.2">first_epsilon_greater_than</a> <span class="p1">(<span class="default"><a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T40" target="_self" title="ORDINAL5:th.40">Th24</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:71_2_2"/>
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font> or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font> or  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">e</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:71_2_2_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:71_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_2"><span class="lab"><font color="Green" title="E50">Z0</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:71_2_2_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:71_2_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_2_3"><span class="kw">suppose </span></a><a NAME="E1:71_2_2_3"/>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]</span><br/><div class="add"><a NAME="E2:71_2_2_3"/><span class="kw">then </span>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_2"><span class="lab"><font color="Green" title="E52">Z2</font></span></a>, <a class="ref" href="ordinal5.html#D6" target="_self" title="ORDINAL5:def.6">FEGT</a></span>;<br/><a NAME="E3:71_2_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_2"><span class="lab"><font color="Green" title="E51">Z1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_2_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:71"/><span class="lab"><font color="Green" title="E50">A2</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>2</sub></font>[<font color="Olive" title="b2">c</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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="c2">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="c2">b</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] ) implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">b</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_3"/><span class="lab"><font color="Green" title="E51">Y0</font></span>: 
( <font color="Maroon" title="c2">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="c2">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E2:71_3"/><span class="lab"><font color="Green" title="E52">Y1</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="c2">b</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">b</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E3:71_3"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]</span><br/>

<a NAME="E4:71_3"/><span class="kw">then </span>
 ex <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 <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Olive" title="b1">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_3"><span class="lab"><font color="Green" title="E51">Y0</font></span></a>, <a class="ref" href="ordinal5.html#T47" target="_self" title="ORDINAL5:th.47">Th38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:71_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_3"><span class="lab"><font color="Green" title="E52">Y1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:71"/><span class="lab"><font color="Green" title="E51">AA</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>  holds  <font color="Maroon">S<sub>2</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="#E1:71"><span class="lab"><font color="Green" title="E48">A0</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E50">A2</font></span></a>);<br/></span>
<a NAME="E5:71"/>
( <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T44" target="_self" title="ORDINAL5:th.44">Th33</a>, <a class="ref" href="ordinal5.html#T48" target="_self" title="ORDINAL5:th.48">Th39</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:71"/>
( <font color="Maroon" title="c1">a</font> is  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  implies  ex <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" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Olive" title="b1">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E51">AA</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
