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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_lar.html#D5" target="_self">CARD_LAR:def 5</a><br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E25</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:31"/><i><font color="Green">E26</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:31_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:31_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="card_lar.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:31_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="card_lar.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E6:31_1"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>
<a NAME="E7:31_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>
<a NAME="E8:31_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:31_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<a NAME="E9:31_1"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="card_lar.html#E7:31_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_lar.html#E1:31"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E5:31"/>
( not <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="ordinal1.html#V4">being_limit_ordinal</a> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:31"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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