<?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"> (  <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">a</font> implies  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:43"/>
 <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">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/>

<a NAME="E2:43"/><span class="kw">then </span>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</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="c1">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/>
<a NAME="E3:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A0</font></span>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or 1 <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><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="E1:43_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E28">A0</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_1_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/><div class="add"><span class="kw">hence </span> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> = 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T63" title="ORDINAL2:th.63">ORDINAL2:63</a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font> <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="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="ordinal5.html#T17" target="_self" title="ORDINAL5:th.17">Th4</a></span>
;<br/> <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="43_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_1_2"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> = <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> $1;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,$1;<br/><span class="kw">consider </span><font color="Maroon" title="c2">T</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:43_1_2"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &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>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  holds <br/><font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></span><span class="kw">consider </span><font color="Maroon" title="c3">E</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E5:43_1_2"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; ( 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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  holds <br/><font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<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#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></span><a NAME="E6:43_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E7:43_1_2"/><span class="kw">then </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">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/><a NAME="E8:43_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font> <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#T23" target="_self" title="ORDINAL5:th.23">Th6</a></span>;<br/><a NAME="E9:43_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
( not <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="ordinal5.html#T30" target="_self" title="ORDINAL5:th.30">Th11</a></span>;<br/><a NAME="E10:43_1_2"/>
<font color="Maroon" title="c3">E</font> is   <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E5:43_1_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="ordinal4.html#T25" title="ORDINAL4:th.25">ORDINAL4:25</a></span>;<br/><a NAME="E11:43_1_2"/><span class="kw">then </span>
(  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">E</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c3">E</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E9:43_1_2"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="ordinal5.html#T6" target="_self" title="ORDINAL5:th.6">ThND</a>, <a class="ref" href="ordinal2.html#T62" title="ORDINAL2:th.62">ORDINAL2:62</a></span>;<br/><a NAME="E12:43_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a></span>;<br/><a NAME="E13:43_1_2"/>
<font color="Maroon" title="c2">T</font> is   <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="ordinal5.html#T25" target="_self" title="ORDINAL5:th.25">Th8</a></span>;<br/><a NAME="E14:43_1_2"/><span class="kw">then </span>
(  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">T</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="ordinal5.html#T15" target="_self" title="ORDINAL5:th.15">Th2</a>, <a class="ref" href="ordinal5.html#T6" target="_self" title="ORDINAL5:th.6">ThND</a></span>;<br/><a NAME="E15:43_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a></span>;<br/><span class="kw">thus </span><a NAME="E16:43_1_2"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
  <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:43_1_2_1"/><span class="lab"><font color="Green" title="E35">B1</font></span>: 
( <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">E</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1_2"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:43_1_2_1"/><span class="lab"><font color="Green" title="E36">B2</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> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> <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="txt" href="#E5:43_1_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E15:43_1_2"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E3:43_1_2_1"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</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="#E3:43_1_2_1"><span class="lab"><font color="Green" title="E35">B1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c6">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E5:43_1_2_1"><span class="lab"><font color="Green" title="E36">B2</font></span></a></span>;<br/>
<a NAME="E8:43_1_2_1"/><span class="lab"><font color="Green" title="E37">B3</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E5:43_1_2_1"><span class="lab"><font color="Green" title="E36">B2</font></span></a>, <a class="ref" href="ordinal4.html#T24" title="ORDINAL4:th.24">ORDINAL4:24</a></span>;<br/>
<a NAME="E9:43_1_2_1"/><span class="lab"><font color="Green" title="E38">B4</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></span>;<br/>
<a NAME="E10:43_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">b</font>) &amp; <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">c</font>) &amp; <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">c</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2_1"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E5:43_1_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a></span>;<br/>
<a NAME="E11:43_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_1_2_1"><span class="lab"><font color="Green" title="E37">B3</font></span></a>, <a class="ref" href="ordinal5.html#T14" target="_self" title="ORDINAL5:th.14">Th1</a></span>;<br/>
<a NAME="E12:43_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2_1"><span class="lab"><font color="Green" title="E35">B1</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:43_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <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="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E15:43_1_2"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E9:43_1_2_1"><span class="lab"><font color="Green" title="E38">B4</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</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><span class="kw">thus </span><a NAME="E17:43_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="ordinal4.html#T32" title="ORDINAL4:th.32">ORDINAL4:32</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><span class="kw">end;</span></div>

</div>
