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

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c1">M</font> is <a href="card_fil.html#V6" title="CARD_FIL:attr.6">strongly_inaccessible</a> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c1">M</font> is <a href="card_fil.html#V5" title="CARD_FIL:attr.5">strong_limit</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> )
 
<b>by </b><i><a class="ref" href="card_fil.html#D15" title="CARD_FIL:def.15">CARD_FIL:def 15</a></i>;<br/>
<a NAME="E3:50"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> $1</span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font> );<br/>
<a NAME="E4:50"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="classes1.html#T33" title="CLASSES1:th.33">CLASSES1:33</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green" title="E42">A5</font></i>: 
for <font color="Olive" title="b1">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B1</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" title="c3">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">B1</font>]
 ;<br/>

<b>assume </b><a NAME="E2:50_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <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>
 ;<br/>

<a NAME="E3:50_1"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:50_1"/><b>then </b><i><font color="Green" title="E44">A7</font></i>: 
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c3">B1</font></span>)</span></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:50"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="card_lar.html#E1:50_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="card_fil.html#D14" title="CARD_FIL:def.14">CARD_FIL:def 14</a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E5:50_1"/>
 <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c3">B1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T34" title="CLASSES1:th.34">CLASSES1:34</a></i>;<br/>
<b>hence </b><a NAME="E6:50_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B1</font></span>)</span></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E4:50_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:50"/><i><font color="Green" title="E43">A8</font></i>: 
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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">B1</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">B2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B2</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B1</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" title="c3">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_2"/>
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:50_2"/><i><font color="Green" title="E44">A9</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>and </b><br/><a NAME="E3:50_2"/><i><font color="Green" title="E45">A10</font></i>: 
for <font color="Olive" title="b1">B2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>]
 ;<br/>

<b>assume </b><a NAME="E4:50_2"/><i><font color="Green" title="E46">A11</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>
 ;<br/>

<a NAME="E5:50_2"/><b>then </b><i><font color="Green" title="E47">A12</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">B1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T25" title="CARD_1:th.25">CARD_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E7:50_2"/><i><font color="Green" title="E48">A13</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B1</font> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> holds <br/><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green" title="E49">A14</font></i>:  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c3">B1</font> = 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c4">L</font>
<b>by </b><i><a class="txt" href="card_lar.html#E2:50_2"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="classes2.html#T25" title="CLASSES2:th.25">CLASSES2:25</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">L</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E9:50_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">B1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="card_1.html#T28" title="CARD_1:th.28">CARD_1:28</a></i>;<br/>
<a NAME="E10:50_2"/><b>then </b><i><font color="Green" title="E50">A15</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:50"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="card_lar.html#E5:50_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E11:50_2"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">L</font> holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">Y</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</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" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_2_2"/><i><font color="Green" title="E51">A16</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">L</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_2_2"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">L</font>
 <b>and </b><br/><a NAME="E4:50_2_2"/><i><font color="Green" title="E53">A18</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:50_2_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">B1</font> <b>by </b><i><a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green" title="E52">A17</font></i></a></i>;<br/>
<a NAME="E6:50_2_2"/><i><font color="Green" title="E54">A19</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:50_2"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E7:50_2_2"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c7">x1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="card_lar.html#E4:50_2_2"><i><font color="Green" title="E53">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:50_2_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">Y</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E3:50_2"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="card_lar.html#E7:50_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="card_lar.html#E3:50_2_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="card_lar.html#E6:50_2_2"><i><font color="Green" title="E54">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:50_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E8:50_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="card_lar.html#E10:50_2"><i><font color="Green" title="E50">A15</font></i></a>, <a class="ref" href="card_lar.html#T34" target="_self" title="CARD_LAR:th.34">Th34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:50"/>
for <font color="Olive" title="b1">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B1</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="card_lar.html#E4:50"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="card_lar.html#E5:50"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="card_lar.html#E6:50"><i><font color="Green" title="E43">A8</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:50"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>


</div>
