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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>defpred </b><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> ] <b>means </b>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="E1:70"/><i><font color="Green" title="E40">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="card_1.html#T46" target="_self" title="CARD_1:th.46">Th46</a></i>;<br/>
<a NAME="E2:70"/><i><font color="Green" title="E41">A2</font></i>: 
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="70_1"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <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="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c3">n</font>]</span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E2:70_1"/><i><font color="Green" title="E43">A4</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E3:70_1"/><b>then </b>
<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> 
 
<b>by </b><i><a class="ref" href="card_1.html#T46" target="_self" title="CARD_1:th.46">Th46</a></i>;<br/>
<a NAME="E4:70_1"/><b>then </b><i><font color="Green" title="E44">A5</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="c4">A</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_1"/><b>then </b><i><font color="Green" title="E45">A6</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="#E4:70_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/><a NAME="E3:70_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">A</font> &amp;  <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> &amp;  <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 <b>by </b><i><a class="txt" href="#E2:70_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="#E2:70_1_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="card_1.html#E45"><i><font color="Green" title="E19">Lm1</font></i></a>, <a class="ref" href="card_1.html#T21" target="_self" title="CARD_1:th.21">Th21</a>, <a class="ref" href="card_1.html#T27" target="_self" title="CARD_1:th.27">Th27</a>, <a class="ref" href="card_1.html#T83" target="_self" title="CARD_1:th.83">Th83</a></i>;<br/><b>hence </b><a NAME="E4:70_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</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>
<b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:70_1"/><i><font color="Green" title="E45">A7</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>
<a NAME="E8:70_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="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> )
 
<b>by </b><i><a class="txt" href="#E7:70_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<a NAME="E9:70_1"/><b>then </b>
( <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>,<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="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &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> &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> )
 
<b>by </b><i><a class="txt" href="#E2:70_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="#E7:70_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="card_1.html#T61" target="_self" title="CARD_1:th.61">Th61</a>, <a class="ref" href="ordinal1.html#T52" title="ORDINAL1:th.52">ORDINAL1:52</a></i>;<br/>
<b>hence </b><a NAME="E10:70_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>
 <b>by </b><i><a class="txt" href="#E1:70_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="#E7:70_1"><i><font color="Green" title="E45">A7</font></i></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="E3:70"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S17" title="ORDINAL2:sch.17">ORDINAL2:sch 17</a>(<a class="txt" href="#E1:70"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="#E2:70"><i><font color="Green" title="E41">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:70"/>
( <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
