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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon" title="c2">X</font> <a href="card_lar.html#R3" title="CARD_LAR:pred.3">is_club_in</a> <font color="Maroon" title="c1">A</font> implies ( <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">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" title="c2">X</font> <a href="card_lar.html#R3" title="CARD_LAR:pred.3">is_club_in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="card_lar.html#R2" title="CARD_LAR:pred.2">is_closed_in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">X</font> <a href="card_lar.html#R1" title="CARD_LAR:pred.1">is_unbounded_in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self" title="CARD_LAR:def.3">Def3</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b>
( ( for <font color="Olive" title="b1">B</font> being <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <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="c1">A</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self" title="CARD_LAR:def.1">Def1</a>, <a class="ref" href="card_lar.html#D2" target="_self" title="CARD_LAR:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1"/>
( <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )
 <b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self" title="CARD_LAR:def.4">Def4</a>, <a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">Def5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:9"/>
( <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )
 ;<br/>

<a NAME="E3:9"/><b>then </b>
( ( for <font color="Olive" title="b1">B</font> being <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <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="c1">A</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self" title="CARD_LAR:def.4">Def4</a>, <a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">Def5</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="card_lar.html#R2" title="CARD_LAR:pred.2">is_closed_in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">X</font> <a href="card_lar.html#R1" title="CARD_LAR:pred.1">is_unbounded_in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self" title="CARD_LAR:def.1">Def1</a>, <a class="ref" href="card_lar.html#D2" target="_self" title="CARD_LAR:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:9"/>
<font color="Maroon" title="c2">X</font> <a href="card_lar.html#R3" title="CARD_LAR:pred.3">is_club_in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self" title="CARD_LAR:def.3">Def3</a></i>;<br/>


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