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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <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; $1 <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">fi</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/><font color="Olive" title="b1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">B</font> ) holds <br/>for <font color="Olive" title="b2">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not <font color="Olive" title="b2">D</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Olive" title="b1">fi</font> ) );<br/>
<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c2">A</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="c2">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">fi</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>  st  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">B</font> ) holds <br/>for <font color="Olive" title="b2">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not <font color="Olive" title="b2">D</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Olive" title="b1">fi</font> ) )
 ;<br/>

<a NAME="E2:27"/><i><font color="Green" title="E25">A2</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:27"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green" title="E27">A4</font></i>: 
for <font color="Olive" title="b1">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font>] holds <br/><font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A1</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="ordinal4.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <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">C</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c4">fi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E7:27"/><i><font color="Green" title="E28">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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">A</font> holds <br/><font color="Maroon" title="c4">fi</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>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<div><i><font color="Green" title="E29">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/>
( <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><a NAME="E2:27_1"/><b>then </b>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c4">fi</font> or 1 <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c4">fi</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="ordinal4.html#E7:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="ordinal4.html#E27"><i><font color="Green" title="E21">Lm6</font></i></a>, <a class="txt" href="ordinal4.html#E28"><i><font color="Green" title="E22">Lm7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="ordinal4.html#E7:27"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:27"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<a NAME="E10:27"/><b>then </b>
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E11:27"/><b>then </b>
1 <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E8:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E12:27"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green" title="E31">A8</font></i>: 
for <font color="Olive" title="b1">B2</font>, <font color="Olive" title="b2">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B2</font> &amp; <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">B1</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">C</font>,<font color="Olive" title="b1">B2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>for <font color="Olive" title="b1">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">B1</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">C</font>,$1;<br/>
<a NAME="E1:27_2"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E2:27_2"/><i><font color="Green" title="E33">A10</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:27_2_1"/><i><font color="Green" title="E34">A11</font></i>: 
for <font color="Olive" title="b1">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">B1</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">C</font>,<font color="Maroon" title="c5">B</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:27_2_1"/><i><font color="Green" title="E35">A12</font></i>: 
( <font color="Maroon" title="c6">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 ;<br/>

<a NAME="E3:27_2_1"/><i><font color="Green" title="E36">A13</font></i>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1_1_1_1"/>
( <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>  or <font color="Maroon" title="c5">B</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_1_1"/>
<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> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_2_1_1_1_1_1"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B</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> 
 <b>by </b><i><a class="ref" href="ordinal2.html#T60" title="ORDINAL2:th.60">ORDINAL2:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_1_2"/>
<font color="Maroon" title="c5">B</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> 
 ;<br/><div class="add"><a NAME="E2:27_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A14</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/><a NAME="E3:27_2_1_1_1_1_2"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><a NAME="E4:27_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E2:27_2_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E5:27_2_1_1_1_1_2"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B</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> 
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:27_2_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="ordinal4.html#E2:27_2_1_1_1_1_2"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_2_1_1"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B</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> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_2_1"/><i><font color="Green" title="E37">A15</font></i>: 
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</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">C</font>,<font color="Maroon" title="c5">B</font></span>)</span> &amp; 1 <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">C</font>,<font color="Maroon" title="c5">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">C</font>,<font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a>, <a class="ref" href="ordinal2.html#T61" title="ORDINAL2:th.61">ORDINAL2:61</a></i>;<br/>
<a NAME="E5:27_2_1"/><b>then </b><i><font color="Green" title="E38">A16</font></i>: 
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font></span>)</span> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">B</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E12:27"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="ordinal4.html#E2:27_2_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="ordinal4.html#E3:27_2_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="ordinal2.html#T57" title="ORDINAL2:th.57">ORDINAL2:57</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_1_2"/>
<font color="Maroon" title="c6">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font>
 ;<br/><a NAME="E2:27_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">B1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:27_2_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:27_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E4:27_2_1_2"/><b>then </b>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B1</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">C</font>,<font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:27_2_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="ordinal4.html#E5:27_2_1"><i><font color="Green" title="E38">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E5:27_2_1_2"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B1</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">C</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:27_2_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_2_1"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B1</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">C</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E12:27"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="ordinal4.html#E3:27_2_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="ordinal4.html#E4:27_2_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="ordinal2.html#T57" title="ORDINAL2:th.57">ORDINAL2:57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_2"/><i><font color="Green" title="E34">A17</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</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">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">D</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_2_2"/><i><font color="Green" title="E35">A18</font></i>: 
( <font color="Maroon" title="c5">B</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="c5">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:27_2_2"/><i><font color="Green" title="E36">A19</font></i>: 
for <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">D</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c6">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E3:27_2_2"/><i><font color="Green" title="E37">A20</font></i>: 
( <font color="Maroon" title="c6">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 ;<br/>

<a NAME="E4:27_2_2"/><b>then </b>
not <font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">psi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:27_2_2"/><i><font color="Green" title="E38">A21</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> &amp; ( for <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/><font color="Maroon" title="c7">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">D</font> ) &amp;  ex <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Olive" title="b1">D</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c7">psi</font> )
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="ordinal4.html#E1:27_2_2"><i><font color="Green" title="E35">A18</font></i></a></i>;<br/>
<a NAME="E7:27_2_2"/><i><font color="Green" title="E39">A22</font></i>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B</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="c7">psi</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:27_2_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="ordinal4.html#E6:27_2_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="ordinal2.html#T62" title="ORDINAL2:th.62">ORDINAL2:62</a></i>;<br/>
<a NAME="E8:27_2_2"/>
<font color="Maroon" title="c7">psi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">B1</font>, <font color="Maroon" title="c9">B2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/>

<b>assume </b><a NAME="E1:27_2_2_1"/><i><font color="Green" title="E40">A23</font></i>: 
( <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B2</font> &amp; <font color="Maroon" title="c9">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">psi</font> )
 ;<br/>

<a NAME="E2:27_2_2_1"/><b>then </b>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">B2</font>] &amp; <font color="Maroon" title="c9">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:27_2_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="ordinal4.html#E3:27_2_2"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="ordinal4.html#E6:27_2_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:27_2_2_1"/><b>then </b>
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c8">B1</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">C</font>,<font color="Maroon" title="c9">B2</font> &amp; <font color="Maroon" title="c7">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon" title="c7">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">B2</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E6:27_2_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="ordinal4.html#E1:27_2_2_1"><i><font color="Green" title="E40">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:27_2_2_1"/>
<font color="Maroon" title="c7">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:27_2_2"/><b>then </b><i><font color="Green" title="E40">A24</font></i>: 
 <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c7">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">psi</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:27_2_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="ordinal4.html#E6:27_2_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="ordinal4.html#T8" target="_self" title="ORDINAL4:th.8">Th8</a></i>;<br/>
<a NAME="E10:27_2_2"/>
<font color="Maroon" title="c7">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B1</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:27_2_2"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="ordinal4.html#E6:27_2_2"><i><font color="Green" title="E38">A21</font></i></a></i>;<br/>
<a NAME="E11:27_2_2"/><b>then </b>
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B1</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="c7">psi</font> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c7">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">psi</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:27_2_2"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="ordinal4.html#E6:27_2_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:27_2_2"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B1</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">C</font>,<font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E7:27_2_2"><i><font color="Green" title="E39">A22</font></i></a>, <a class="txt" href="ordinal4.html#E9:27_2_2"><i><font color="Green" title="E40">A24</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:27_2"/>
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 <b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal4.html#E1:27_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="ordinal4.html#E2:27_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="ordinal4.html#E3:27_2"><i><font color="Green" title="E34">A17</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E14:27"/>
<font color="Maroon" title="c4">fi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B1</font>, <font color="Maroon" title="c6">B2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a><br/>

<b>assume </b><a NAME="E1:27_3"/><i><font color="Green" title="E32">A25</font></i>: 
( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B2</font> &amp; <font color="Maroon" title="c6">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">fi</font> )
 ;<br/>

<a NAME="E2:27_3"/><b>then </b>
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">fi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:27_3"/><b>then </b>
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B1</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">C</font>,<font color="Maroon" title="c6">B2</font> &amp; <font color="Maroon" title="c4">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">B1</font> &amp; <font color="Maroon" title="c4">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B2</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E7:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="ordinal4.html#E13:27"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="ordinal4.html#E1:27_3"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:27_3"/>
<font color="Maroon" title="c4">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c4">fi</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c4">fi</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="ordinal4.html#E7:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="ordinal4.html#T8" target="_self" title="ORDINAL4:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E16:27"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="ordinal4.html#E7:27"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


</div>
