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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</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">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>  st  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b1">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> holds <br/> <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Olive" title="b1">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a></span><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">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> implies  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )</span><br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> or  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )</span><br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a></span><br/>

<a NAME="E3:39"/>
for <font color="Olive" title="b1">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/> ex <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">B1</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="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> implies  ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> ) )</span><br/>

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">B</font> being  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">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><b> such that </b><br/><a NAME="E3:39_1"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E4:39_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E5:39_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:39"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E1:39_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="card_lar.html#T25" target="_self" title="CARD_LAR:th.25">Th25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font> )</span><br/>

<b>thus </b><a NAME="E6:39_1"/>
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="#E4:39_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E5:39_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</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>hence </b><a NAME="E4:39"/>
 <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <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#T7" target="_self" title="CARD_LAR:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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