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

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:3_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
not <font color="Maroon" title="c1">M</font> is <a href="card_1.html#V2" title="CARD_1:attr.2">limit</a>
 ;<br/>

<b>assume </b><a NAME="E2:3_1_1"/>
<font color="Maroon" title="c1">M</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 ;<br/>

<a NAME="E3:3_1_1"/><b>then </b><i><font color="Green" title="E2">A2</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#D1" title="CARD_4:def.1">CARD_4:def 1</a></i>;<br/>
<a NAME="E4:3_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 ;<br/>

<a NAME="E2:3_1_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:3_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="card_4.html#T2" title="CARD_4:th.2">CARD_4:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:3_1_1"/><b>then </b>
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T83" title="CARD_1:th.83">CARD_1:83</a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E1:3_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="card_1.html#T85" title="CARD_1:th.85">CARD_1:85</a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>


</div>
