<?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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &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  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E10">A0</font></span>: 
( <font color="Maroon" title="c1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &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="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>,<font color="Maroon" title="c2">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_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="E1:18_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <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> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><span class="kw">suppose </span></a><a NAME="E1:18_1_1"/>
<font color="Maroon" title="c1">a</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="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>,<font color="Maroon" title="c2">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:18_1_1"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> 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="#E1:18"><span class="lab"><font color="Green" title="E10">A0</font></span></a>, <a class="ref" href="ordinal4.html#T20" title="ORDINAL4:th.20">ORDINAL4:20</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="18_1_2"><span class="kw">suppose </span></a><a NAME="E1:18_1_2"/><span class="lab"><font color="Green" title="E11">0A</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>
 ; <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>,<font color="Maroon" title="c2">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> </span><br/><div class="add"><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  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,$1 is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  );<br/><a NAME="E2:18_1_2"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/><a NAME="E3:18_1_2"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
for <font color="Olive" title="b1">c</font> being  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">c</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><span class="kw">proof </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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">c</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">c</font>] )</span><br/>

<a NAME="E1:18_1_2_1"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T61" title="ORDINAL2:th.61">ORDINAL2:61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:18_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">c</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">c</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E10">A0</font></span></a>, <a class="ref" href="ordinal3.html#T48" title="ORDINAL3:th.48">ORDINAL3:48</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="E4:18_1_2"/><span class="lab"><font color="Green" title="E14">A3</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#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">c</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2"><span class="kw">proof </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="Red">::  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><span class="kw">that </span><br/><a NAME="E1:18_1_2_2"/><span class="lab"><font color="Green" title="E15">Z0</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>  )
 <span class="kw">and </span><br/><a NAME="E2:18_1_2_2"/><span class="lab"><font color="Green" title="E16">Z1</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#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="Red">::  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="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="c4">f</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E4:18_1_2_2"/><span class="lab"><font color="Green" title="E17">B1</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</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">f</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="E5:18_1_2_2"/><span class="lab"><font color="Green" title="E18">B2</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_2_2"><span class="lab"><font color="Green" title="E15">Z0</font></span></a>, <a class="txt" href="#E4:18_1_2_2"><span class="lab"><font color="Green" title="E17">B1</font></span></a>, <a class="ref" href="ordinal2.html#T62" title="ORDINAL2:th.62">ORDINAL2:62</a></span>;<br/>
<a NAME="E6:18_1_2_2"/>
<font color="Maroon" title="c4">f</font> is  <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E11">0A</font></span></a>, <a class="txt" href="#E4:18_1_2_2"><span class="lab"><font color="Green" title="E17">B1</font></span></a>, <a class="ref" href="ordinal5.html#T8" target="_self" title="ORDINAL5:th.8">Th003</a></span>;<br/>
<a NAME="E7:18_1_2_2"/><span class="kw">then </span>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c4">f</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_2_2"><span class="lab"><font color="Green" title="E15">Z0</font></span></a>, <a class="txt" href="#E4:18_1_2_2"><span class="lab"><font color="Green" title="E17">B1</font></span></a>, <a class="ref" href="ordinal5.html#T6" target="_self" title="ORDINAL5:th.6">ThND</a></span>;<br/>
<a NAME="E8:18_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">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="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">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_2_2"><span class="lab"><font color="Green" title="E18">B2</font></span></a>, <a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a></span>;<br/>
<a NAME="E9:18_1_2_2"/>
for <font color="Olive" title="b1">d</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">d</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>,<font color="Maroon" title="c3">c</font> holds <br/> <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">d</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>,<font color="Maroon" title="c3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">d</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="c5">d</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>,<font color="Maroon" title="c3">c</font> implies  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">d</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>,<font color="Maroon" title="c3">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1_2_2_1"/>
<font color="Maroon" title="c5">d</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>,<font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">d</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>,<font color="Maroon" title="c3">c</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:18_1_2_2_1"/><span class="lab"><font color="Green" title="E20">B4</font></span>: 
( <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">f</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</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="#E8:18_1_2_2"><span class="lab"><font color="Green" title="E19">B3</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="c6">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="#E3:18_1_2_2_1"><span class="lab"><font color="Green" title="E20">B4</font></span></a></span>;<br/>
<a NAME="E5:18_1_2_2_1"/><span class="lab"><font color="Green" title="E21">B5</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">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">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_2_2"><span class="lab"><font color="Green" title="E15">Z0</font></span></a>, <a class="txt" href="#E4:18_1_2_2"><span class="lab"><font color="Green" title="E17">B1</font></span></a>, <a class="txt" href="#E3:18_1_2_2_1"><span class="lab"><font color="Green" title="E20">B4</font></span></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></span>;<br/>
<a NAME="E6:18_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">B6</font></span>: 
( <font color="Maroon" title="c4">f</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>1</sub></font>(<font color="Maroon" title="c7">b</font>) &amp; <font color="Maroon" title="c4">f</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="c7">b</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="c7">b</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">b</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_2_2"><span class="lab"><font color="Green" title="E16">Z1</font></span></a>, <a class="txt" href="#E4:18_1_2_2"><span class="lab"><font color="Green" title="E17">B1</font></span></a>, <a class="txt" href="#E3:18_1_2_2_1"><span class="lab"><font color="Green" title="E20">B4</font></span></a></span>;<br/>
<a NAME="E7:18_1_2_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">b</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E11">0A</font></span></a>, <a class="ref" href="ordinal4.html#T27" title="ORDINAL4:th.27">ORDINAL4:27</a>, <a class="ref" href="ordinal3.html#T1" title="ORDINAL3:th.1">ORDINAL3:1</a></span>;<br/>
<a NAME="E8:18_1_2_2_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</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="c7">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_1_2_2_1"><span class="lab"><font color="Green" title="E22">B6</font></span></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a>, <a class="txt" href="#E3:18_1_2_2_1"><span class="lab"><font color="Green" title="E20">B4</font></span></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:18_1_2_2_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">d</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>,<font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18_1_2_2"><span class="lab"><font color="Green" title="E19">B3</font></span></a>, <a class="txt" href="#E5:18_1_2_2_1"><span class="lab"><font color="Green" title="E21">B5</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">hence </span><a NAME="E10:18_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">c</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</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="E5:18_1_2"/>
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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</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="#E2:18_1_2"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:18_1_2"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E4:18_1_2"><span class="lab"><font color="Green" title="E14">A3</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E6:18_1_2"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> 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="#E1:18"><span class="lab"><font color="Green" title="E10">A0</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><span class="kw">end;</span></div>

</div>
