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

<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  holds <br/><font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  implies <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <span class="kw">means </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>,$1 <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<a NAME="E1:69"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</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="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">A</font>, <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  implies <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<a NAME="E2:69_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal1.html#T37" title="ORDINAL1:th.37">ORDINAL1:37</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c4">A</font>, <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</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="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E4:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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">by</span> <span class="lab"><a class="ref" href="card_1.html#T26" target="_self" title="CARD_1:th.26">Th46</a></span>;<br/>
<a NAME="E5:69_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A5</font></span>: 
 <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="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:69_1_1"/>
<font color="Maroon" title="c4">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:69_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A6</font></span>: 
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="ordinal1.html#D11" title="ORDINAL1:def.11">ORDINAL1:def 11</a></span>;<br/><a NAME="E3:69_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="card_1.html#T5" target="_self" title="CARD_1:th.5">Th21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:69_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="card_1.html#E24"><span class="lab"><font color="Green" title="E20">Lm1</font></span></a>, <a class="ref" href="card_1.html#T11" target="_self" title="CARD_1:th.11">Th27</a>, <a class="ref" href="card_1.html#T46" target="_self" title="CARD_1:th.46">Th83</a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E8:69_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T29" title="ORDINAL1:th.29">ORDINAL1:29</a></span>;<br/>
<a NAME="E9:69_1"/>
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a>, <a class="ref" href="ordinal1.html#T37" title="ORDINAL1:th.37">ORDINAL1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:69_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:69_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E2:69_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="card_1.html#T34" target="_self" title="CARD_1:th.34">Th61</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="E2:69"/><span class="lab"><font color="Green" title="E44">A8</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="card_1.html#T26" target="_self" title="CARD_1:th.26">Th46</a></span>;<br/>
<a NAME="E3:69"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S17" title="ORDINAL2:sch.17">ORDINAL2:sch 17</a>(<a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E43">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:69"/>
( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  implies <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</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/>


</div>
