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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <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="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">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> <font color="Maroon" title="c2">b</font> implies <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> <font color="Maroon" title="c2">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E21">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">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> <font color="Maroon" title="c2">b</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="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> <font color="Maroon" title="c2">b</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>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 implies <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> $1 );<br/>
<a NAME="E2:33"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><span class="lab"><font color="Green" title="E23">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">now </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/><span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<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"> <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="E2:33_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon" title="c1">a</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> <font color="Maroon" title="c3">b</font></span>)</span> <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> <font color="Maroon" title="c3">b</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 NAME="E3:33_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><span class="kw">thus </span><a NAME="E4:33_1"/>
<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>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_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="E1:33_1_1_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="c3">b</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_1_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="c3">b</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">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/><div class="add"><a NAME="E2:33_1_1_1_1"/><span class="kw">then </span>
1 <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> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E3:33_1_1_1_1"/><span class="kw">then </span>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">a</font>,1) <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> <font color="Maroon" title="c3">b</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="ordinal4.html#T27" title="ORDINAL4:th.27">ORDINAL4:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:33_1_1_1_1"/>
<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="#E2:33_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="ordinal2.html#T46" title="ORDINAL2:th.46">ORDINAL2:46</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="hsNdiv(this)" href="javascript:()" title="33_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">b</font>]</span><br/><div class="add"><a NAME="E2:33_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T13" target="_self" title="ORDINAL5:th.13">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_1_1_2"/>
<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="#E2:33_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="ordinal2.html#T46" title="ORDINAL2:th.46">ORDINAL2:46</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E24">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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="c3">c</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">c</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</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="c3">c</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">c</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:33_2"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
( <font color="Maroon" title="c3">c</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">c</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</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="c3">c</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">c</font>]</span><br/><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">consider </span><font color="Maroon" title="c4">phi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E3:33_2"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &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="c3">c</font> holds <br/><font color="Maroon" title="c4">phi</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>1</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="E4:33_2"/>
<font color="Maroon" title="c4">phi</font> is  <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">e</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="ordinal5.html#D2" target="_self" title="ORDINAL5:def.2">ORDINAL5:def 2</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">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="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> &amp; <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> <font color="Maroon" title="c4">phi</font> holds <br/><font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">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="c4">phi</font> implies <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:33_2_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">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="c4">phi</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="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span><br/>

<a NAME="E2:33_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A11</font></span>: 
( <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">b</font>) &amp; <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E3:33_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">e</font>) &amp; <font color="Maroon" title="c5">e</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E1:33_2_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:33_2_1"/>
<font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E2:33_2_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="ordinal5.html#T21" target="_self" title="ORDINAL5:th.21">Th21</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="E5:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c4">phi</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c4">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="ordinal5.html#T6" target="_self" title="ORDINAL5:th.6">Th6</a></span>;<br/><a NAME="E6:33_2"/>
 <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="ordinal5.html#T15" target="_self" title="ORDINAL5:th.15">Th15</a></span>;<br/><a NAME="E7:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>) <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="c4">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="ordinal2.html#D10" title="ORDINAL2:def.10">ORDINAL2:def 10</a></span>;<br/><span class="kw">thus </span><a NAME="E8:33_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">c</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:33_2_2"/>
 <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="c3">c</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="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> <font color="Maroon" title="c3">c</font></span><br/>

<a NAME="E2:33_2_2"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <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="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="ordinal1.html#T28" title="ORDINAL1:th.28">ORDINAL1:28</a></span>;<br/>
<a NAME="E3:33_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="c4">phi</font> &amp; <font color="Maroon" title="c4">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(1) &amp; <font color="Maroon">H<sub>1</sub></font>(1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="ordinal5.html#T16" target="_self" title="ORDINAL5:th.16">Th16</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="E4:33_2_2"/>
<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> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_2"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</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><span class="kw">end;</span></div>
<a NAME="E5:33"/>
 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="#E2:33"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A7</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:33"/>
<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> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E21">A1</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>
