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

<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">B</font></span>)</span> <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="c2">A</font></span>)</span></span><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> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,$1</span>)</span> <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="c2">A</font></span>)</span>;<br/>
<a NAME="E1:37"/>
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c2">A</font></span>)</span> &amp; 1 <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>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a>, <a class="ref" href="ordinal2.html#T60" title="ORDINAL2:th.60">ORDINAL2:60</a></i>;<br/>
<a NAME="E2:37"/><b>then </b><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green" title="E31">A2</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>1</sub></font>[<font color="Olive" title="b1">B</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">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">B</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">B</font>] )</span><br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E32">A3</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"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">B</font></span>)</span> <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="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">B</font>]</span><br/>

<b>thus </b> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> = 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></i>
<br/>.= 
<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>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T61" title="ORDINAL2:th.61">ORDINAL2:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p2">(<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="c4">B</font></span>)</span></span>)</span> <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="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:37_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="ordinal3.html#T58" title="ORDINAL3:th.58">ORDINAL3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <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="c2">A</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T61" title="ORDINAL2:th.61">ORDINAL2:61</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:37"/><i><font color="Green" title="E32">A4</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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">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>1</sub></font>[<font color="Olive" title="b2">D</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c4">B</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> &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="c4">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">D</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">B</font>] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_2"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c4">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="c4">B</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> )
 <b>and </b><br/><a NAME="E2:37_2"/><i><font color="Green" title="E34">A6</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="c4">B</font> holds <br/> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b1">D</font></span>)</span> <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="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">B</font>]</span><br/>

<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="c5">fi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:37_2"/><i><font color="Green" title="E35">A7</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">B</font> holds <br/><font color="Maroon" title="c5">fi</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c6">psi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:37_2"/><i><font color="Green" title="E36">A8</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">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="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Maroon" title="c6">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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E7:37_2"/>
( <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">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="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> )
 
<b>by </b><i><a class="txt" href="#E1:37_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="ordinal3.html#T29" title="ORDINAL3:th.29">ORDINAL3:29</a>, <a class="ref" href="ordinal3.html#T32" title="ORDINAL3:th.32">ORDINAL3:32</a></i>;<br/>
<a NAME="E8:37_2"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
 <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c6">psi</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:37_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="ordinal2.html#T62" title="ORDINAL2:th.62">ORDINAL2:62</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</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="c7">f1</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E10:37_2"/><i><font color="Green" title="E38">A10</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</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="c2">A</font> holds <br/><font color="Maroon" title="c7">f1</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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E11:37_2"/><i><font color="Green" title="E39">A11</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <span class="p3">(<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="c2">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E6:37_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="#E10:37_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/>
<div><i><font color="Green" title="E40">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">D</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">D</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="c7">f1</font> implies <font color="Maroon" title="c6">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font> )</span><br/><b>assume </b><a NAME="E1:37_2_1"/><i><font color="Green" title="E41">A13</font></i>: 
<font color="Maroon" title="c8">D</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="c7">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font></span><br/><a NAME="E2:37_2_1"/>
<font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></i>;<br/><b>hence </b><font color="Maroon" title="c6">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font> = 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c8">D</font>
<b>by </b><i><a class="txt" href="#E6:37_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="#E10:37_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="#E1:37_2_1"><i><font color="Green" title="E41">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font>
<b>by </b><i><a class="txt" href="#E10:37_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="#E1:37_2_1"><i><font color="Green" title="E41">A13</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E13:37_2"/><i><font color="Green" title="E41">A14</font></i>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">fi</font>
 
<b>by </b><i><a class="txt" href="#E1:37_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="ordinal4.html#T21" target="_self" title="ORDINAL4:th.21">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">D</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">D</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <span class="p2">(<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="c2">A</font></span>)</span></span>)</span> implies <font color="Maroon" title="c6">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <span class="p2">(<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="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font> )</span><br/><b>assume </b><a NAME="E1:37_2_2"/>
<font color="Maroon" title="c8">D</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <span class="p2">(<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="c2">A</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <span class="p2">(<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="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font></span><br/><a NAME="E2:37_2_2"/><b>then </b><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c8">D</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">fi</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>;<br/><a NAME="E3:37_2_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="ordinal2.html#T49" title="ORDINAL2:th.49">ORDINAL2:49</a></i>;<br/><b>hence </b><font color="Maroon" title="c6">psi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">D</font></span>)</span> = 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">D</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:37_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="#E10:37_2"><i><font color="Green" title="E38">A10</font></i></a></i>
<br/>.= 
<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="c8">D</font></span>)</span> <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="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:37_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E2:37_2_2"><i><font color="Green" title="E42">A15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font></span>)</span> <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="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="#E2:37_2_2"><i><font color="Green" title="E42">A15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <span class="p2">(<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="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font>
<b>by </b><i><a class="txt" href="#E2:37_2_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="ordinal3.html#D5" title="ORDINAL3:def.5">ORDINAL3:def 5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E15:37_2"/><b>then </b>
( <font color="Maroon" title="c6">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</a> <span class="p2">(<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="c2">A</font></span>)</span></span>)</span> &amp; <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="c4">B</font></span>)</span> <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="c2">A</font></span>)</span> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c5">fi</font> <a href="ordinal3.html#K4" title="ORDINAL3:func.4">*^</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="c2">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E11:37_2"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="#E12:37_2"><i><font color="Green" title="E40">A12</font></i></a>, <a class="txt" href="#E13:37_2"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="ordinal4.html#D1" target="_self" title="ORDINAL4:def.1">Def1</a>, <a class="ref" href="ordinal4.html#T5" target="_self" title="ORDINAL4:th.5">Th5</a></i>;<br/>
<a NAME="E16:37_2"/><b>then </b>
<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="c4">B</font></span>)</span> <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="c2">A</font></span>)</span> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c6">psi</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T3" target="_self" title="ORDINAL4:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E17:37_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">B</font>]
 <b>by </b><i><a class="txt" href="#E8:37_2"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="ordinal2.html#D14" title="ORDINAL2:def.14">ORDINAL2:def 14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:37"/>
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>1</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="#E2:37"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E3:37"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="#E4:37"><i><font color="Green" title="E32">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:37"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">B</font></span>)</span> <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="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
