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

<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>thus </b><a NAME="E1:9"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="card_lar.html#R3" target="_self">is_club_in</a> <font color="Maroon">c<sub>1</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font> is <a href="card_lar.html#V2" target="_self">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="card_lar.html#R3" target="_self">is_club_in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="card_lar.html#R2" target="_self">is_closed_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="card_lar.html#R1" target="_self">is_unbounded_in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b>
( ( for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &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="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp;  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self">Def1</a>, <a class="ref" href="card_lar.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="card_lar.html#V2" target="_self">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a> )
 <b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self">Def4</a>, <a class="ref" href="card_lar.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:9"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="card_lar.html#V2" target="_self">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a> )
 ;<br/>

<a NAME="E3:9"/><b>then </b>
( ( for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &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="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp;  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self">Def4</a>, <a class="ref" href="card_lar.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="card_lar.html#R2" target="_self">is_closed_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="card_lar.html#R1" target="_self">is_unbounded_in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self">Def1</a>, <a class="ref" href="card_lar.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:9"/>
<font color="Maroon">c<sub>2</sub></font> <a href="card_lar.html#R3" target="_self">is_club_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self">Def3</a></i>;<br/>


</div>
