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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:52"/><i><font color="Green">E35</font></i>: 
 <a href="card_fil.html#R3" target="_self">GCH</a> 
 ;<br/>

<b>assume </b><a NAME="E2:52"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="card_1.html#V2">limit</a>
 ;<br/>

<b>assume </b><a NAME="E3:52"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="card_fil.html#V5" target="_self">strong_limit</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="card_1.html#NM1">Cardinal</a><b> such that </b><br/><a NAME="E5:52"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font> &amp; not  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="card_fil.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E6:52"/><i><font color="Green">E38</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:52_1"/><b>then </b>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,2 is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="fraenkel.html#T16">FRAENKEL:16</a></i>;<br/>
<a NAME="E3:52_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,2</span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/>
<a NAME="E4:52_1"/><b>then </b>
 <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_2.html#D3">CARD_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E5:52"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_4.html#T13">CARD_4:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:52"/>
<font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:52"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>
<a NAME="E8:52"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:52"><i><font color="Green">E35</font></i></a>, <a class="txt" href="card_fil.html#E6:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="card_fil.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E9:52"/>
 <a href="card_1.html#K2">nextcard</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:52"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_4.html#T22">CARD_4:22</a></i>;<br/>
<a NAME="E10:52"/><b>then </b>
 <a href="card_1.html#K2">nextcard</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="txt" href="card_fil.html#E8:52"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:52"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E2:52"><i><font color="Green">E36</font></i></a>, <a class="ref" href="card_1.html#D7">CARD_1:def 7</a></i>;<br/>


</div>
