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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="card_4.html#V1">countable</a> <a href="card_5.html#NM1">Aleph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="card_fil.html#V6">strongly_inaccessible</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="card_fil.html#V5">strong_limit</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="card_5.html#V1">regular</a> )
 
<b>by </b><i><a class="ref" href="card_fil.html#D15">CARD_FIL:def 15</a></i>;<br/>
<a NAME="E3:50"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</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_5.html#D4">CARD_5:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<a NAME="E4:50"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="classes1.html#T33">CLASSES1:33</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:50_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:50_1"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:50_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:50"><i><font color="Green">E39</font></i></a>, <a class="txt" href="card_lar.html#E1:50_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_fil.html#D14">CARD_FIL:def 14</a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E5:50_1"/>
 <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T34">CLASSES1:34</a></i>;<br/>
<b>hence </b><a NAME="E6:50_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E4:50_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_2.html#T44">CARD_2:44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:50"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:50_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>and </b><br/><a NAME="E3:50_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <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>3</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E4:50_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E5:50_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</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_1.html#T25">CARD_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E7:50_2"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <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>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green">E49</font></i>:  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="card_lar.html#E2:50_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="classes2.html#T25">CLASSES2:25</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E9:50_2"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_1.html#T28">CARD_1:28</a></i>;<br/>
<a NAME="E10:50_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:50"><i><font color="Green">E40</font></i></a>, <a class="txt" href="card_lar.html#E5:50_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E11:50_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_2_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:50_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:50_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E6:50_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:50_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E7:50_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_lar.html#E4:50_2_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:50_2_2"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</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="txt" href="card_lar.html#E3:50_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_lar.html#E6:50_2_2"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:50_2"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E8:50_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="card_lar.html#E10:50_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_lar.html#T34" target="_self">Th34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:50"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="card_lar.html#E4:50"><i><font color="Green">E41</font></i></a>, <a class="txt" href="card_lar.html#E5:50"><i><font color="Green">E42</font></i></a>, <a class="txt" href="card_lar.html#E6:50"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:50"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:50"><i><font color="Green">E38</font></i></a></i>;<br/>


</div>
