<?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>, <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="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</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="c3">c</font> implies <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</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><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E21">A2</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="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E3:31"/>
(  <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="c3">c</font> &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="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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/>
<a NAME="E4:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">AA</font></span>: 
( 1 <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">c</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</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="31_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="E1:31_1"/>
( <font color="Maroon" title="c3">c</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="c3">c</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E22">AA</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="31_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c3">c</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"> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">b</font></span><br/><div class="add"><a NAME="E2:31_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">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#T17" target="_self" title="ORDINAL5:th.17">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2"/><span class="lab"><font color="Green" title="E23">A0</font></span>: 
1 <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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c2">b</font></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>for <font color="Olive" title="b1">a</font>, <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="b1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> $1 holds <br/><font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Olive" title="b1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Olive" title="b2">b</font>;<br/><a NAME="E2:31_1_2"/><span class="lab"><font color="Green" title="E24">P0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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="c4">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:31_1_2_1"/><span class="lab"><font color="Green" title="E25">B0</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:31_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_1_2_1"/>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_1"><span class="lab"><font color="Green" title="E25">B0</font></span></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</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="lab"><font color="Green" title="E25">P1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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">S<sub>1</sub></font>[<font color="Maroon" title="c4">d</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="c4">d</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:31_1_2_2"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">d</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>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font>]</span><br/><a NAME="E2:31_1_2_2"/>
<font color="Maroon" title="c3">c</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="c4">d</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="c3">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T14" target="_self" title="ORDINAL5:th.14">Th1</a></span>;<br/><a NAME="E3:31_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">d</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</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="c4">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2"><span class="lab"><font color="Green" title="E23">A0</font></span></a>, <a class="ref" href="ordinal4.html#T32" title="ORDINAL4:th.32">ORDINAL4:32</a></span>;<br/><span class="kw">thus </span><a NAME="E4:31_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font> implies <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:31_1_2_2_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font></span><br/>

<a NAME="E2:31_1_2_2_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_2_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_2_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="E1:31_1_2_2_1_1"/>
( <font color="Maroon" title="c6">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">d</font> or ( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font> ) or ( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">d</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_2_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E2:31_1_2_2_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="ordinal5.html#T1" target="_self" title="ORDINAL5:th.1">Th001</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_2_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_2"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E1:31_1_2_2_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_2_1_1_2"/>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font></span><br/><div class="add"><a NAME="E2:31_1_2_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_2"><span class="lab"><font color="Green" title="E26">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_1_2_2_1_1_3"/>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c5">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_2_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E1:31_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E26">P2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="c4">d</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="c4">d</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">d</font>] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_1_2_3"/><span class="lab"><font color="Green" title="E27">B1</font></span>: 
( <font color="Maroon" title="c4">d</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="c4">d</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E2:31_1_2_3"/><span class="lab"><font color="Green" title="E28">B2</font></span>: 
for <font color="Olive" title="b1">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">d</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="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> $1;<br/><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E4:31_1_2_3"/><span class="lab"><font color="Green" title="E29">B3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; ( for <font color="Olive" title="b1">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e</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="E5:31_1_2_3"/>
<font color="Maroon" title="c5">f</font> is  <a href="ordinal5.html#V2" title="ORDINAL5:attr.2">non-decreasing</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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> ; <span class="comment"><font color="Red">:: 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="Red">::  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="c6">a</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="c5">f</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</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="c7">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="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b</font> &amp; <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="c5">f</font> implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:31_1_2_3_1"/><span class="lab"><font color="Green" title="E30">C0</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b</font> &amp; <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="c5">f</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" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span><br/>

<a NAME="E2:31_1_2_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">b</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_2_3"><span class="lab"><font color="Green" title="E28">B2</font></span></a>, <a class="txt" href="#E4:31_1_2_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E3:31_1_2_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">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>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1_2_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="txt" href="#E1:31_1_2_3_1"><span class="lab"><font color="Green" title="E30">C0</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_1_2_3_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1_2_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:31_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B4</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c5">f</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_3"><span class="lab"><font color="Green" title="E27">B1</font></span></a>, <a class="txt" href="#E4:31_1_2_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="ref" href="ordinal5.html#T6" target="_self" title="ORDINAL5:th.6">ThND</a></span>;<br/><a NAME="E7:31_1_2_3"/>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">d</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="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_3"><span class="lab"><font color="Green" title="E27">B1</font></span></a>, <a class="txt" href="#E4:31_1_2_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="ref" href="ordinal5.html#T15" target="_self" title="ORDINAL5:th.15">Th2</a></span>;<br/><a NAME="E8:31_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B5</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c4">d</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="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_1_2_3"><span class="lab"><font color="Green" title="E30">B4</font></span></a>, <a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a></span>;<br/><span class="kw">thus </span><a NAME="E9:31_1_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">d</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">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="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">d</font> implies <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c7">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:31_1_2_3_2"/><span class="lab"><font color="Green" title="E32">C1</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c7">b</font></span><br/>

<a NAME="E2:31_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">C2</font></span>: 
( ( <font color="Maroon" title="c7">b</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">d</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> ) &amp; ( <font color="Maroon" title="c6">a</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</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="31_1_2_3_2_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="E1:31_1_2_3_2_1"/>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> or ( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> ) or ( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_2_3_2"><span class="lab"><font color="Green" title="E33">C2</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="31_1_2_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_3_2_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c7">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_3_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">a</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2_3_2"><span class="lab"><font color="Green" title="E32">C1</font></span></a>, <a class="txt" href="#E2:31_1_2_3"><span class="lab"><font color="Green" title="E28">B2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E34">C3</font></span>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c7">b</font></span><br/><div class="add"><a NAME="E2:31_1_2_3_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <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="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">a</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1_2_3"><span class="lab"><font color="Green" title="E29">B3</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_1_2_3_2_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">a</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31_1_2_3"><span class="lab"><font color="Green" title="E31">B5</font></span></a>, <a class="txt" href="#E1:31_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E34">C3</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</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="31_1_2_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_3_2_1_3"/>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</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" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c6">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c7">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_3_2_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">a</font>) <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:31_1_2"/>
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="#E2:31_1_2"><span class="lab"><font color="Green" title="E24">P0</font></span></a>, <a class="txt" href="#E3:31_1_2"><span class="lab"><font color="Green" title="E25">P1</font></span></a>, <a class="txt" href="#E4:31_1_2"><span class="lab"><font color="Green" title="E26">P2</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E6:31_1_2"/>
<font color="Maroon" title="c3">c</font> <a href="ordinal5.html#K1" title="ORDINAL5:func.1">|^|^</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">c</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="#E1:31"><span class="lab"><font color="Green" title="E20">A1</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>
