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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</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">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  <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">n</font> holds <br/><font color="Olive" title="b1">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span><br/><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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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">n</font> implies <font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E36">A1</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">n</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="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span><br/>

<a NAME="E2:57"/>
 <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#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E3:57"/><span class="kw">then </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">e</font> &amp; <font color="Maroon" title="c2">e</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="ordinal5.html#T23" target="_self" title="ORDINAL5:th.23">Th23</a>, <a class="ref" href="ordinal5.html#T37" target="_self" title="ORDINAL5:th.37">Th37</a></span>;<br/>
<a NAME="E4:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T14" target="_self" title="ORDINAL5:th.14">Th14</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<font color="Maroon" title="c2">e</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#D5" target="_self" title="ORDINAL5:def.5">Def5</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">e</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T31" title="ORDINAL4:th.31">ORDINAL4:31</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">e</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">e</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">e</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T14" target="_self" title="ORDINAL5:th.14">Th14</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">e</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">e</font>,1)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T46" title="ORDINAL2:th.46">ORDINAL2:46</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">e</font>,<span class="p2">(<span class="default">1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T30" title="ORDINAL4:th.30">ORDINAL4:30</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c2">e</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="card_2.html#T74" title="CARD_2:th.74">CARD_2:74</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T14" target="_self" title="ORDINAL5:th.14">Th14</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</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>
