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

<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font>) )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>$1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,$1);<br/>
<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font>)</span><br/>

<a NAME="E2:40"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 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="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">B</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">B</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B</font>]</span><br/>

<a NAME="E2:40_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <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>) <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="c3">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T39" title="ORDINAL2:th.39">ORDINAL2:39</a></span>;<br/>
<a NAME="E3:40_1"/>
 <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="c3">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="c3">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></span>;<br/>
<a NAME="E4:40_1"/><span class="kw">then </span>
 <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>) <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="c3">B</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="ordinal4.html#T22" target="_self" title="ORDINAL4:th.22">Th22</a>, <a class="ref" href="ordinal2.html#T40" title="ORDINAL2:th.40">ORDINAL2:40</a></span>;<br/>
<a NAME="E5:40_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c3">B</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:40"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">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="Olive" title="b1">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#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="Olive" title="b1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,$1);<br/>
<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="c3">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#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">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c3">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> 
 <span class="kw">and </span><br/><a NAME="E2:40_2"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">and </span><br/><a NAME="E3:40_2"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
 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="Olive" title="b1">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">B</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">fi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E5:40_2"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>) ) )
 <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>
<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c5">x</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="c3">A</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E6:40_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</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="c3">A</font>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E8:40_2"/>
<font color="Maroon" title="c4">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E6:40_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a></span>;<br/>
<a NAME="E9:40_2"/><span class="kw">then </span>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B</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="c4">fi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E6:40_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A11</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c4">fi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E11:40_2"/>
<font color="Maroon" title="c4">fi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="ordinal4.html#T25" target="_self" title="ORDINAL4:th.25">Th25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A12</font></span>:  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c4">fi</font> = 
 <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c4">fi</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="ordinal4.html#T8" target="_self" title="ORDINAL4:th.8">Th8</a></span>
<br/>.= 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">A</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></span>
;<br/>
<a NAME="E13:40_2"/>
<font color="Maroon" title="c6">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E6:40_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:40_2"/>
<font color="Maroon" title="c5">x</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="c3">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E10:40_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <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#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<a NAME="E5:40"/>
 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>]
 
<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="#E4:40"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E33">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:40"/>
<font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">A</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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