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<div class="add">

<b>let </b><font color="Maroon">M</font> be  non <a href="card_4.html#V1">countable</a> <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:51"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">M</font> is <a href="card_fil.html#V6">strongly_inaccessible</a>
 ;<br/>

<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:51"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green">E44</font></i>:  <a href="classes1.html#K4">Rank</a> <font color="Maroon">M</font> = 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="card_lar.html#T3" target="_self">Th3</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">L</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E5:51"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T44">CLASSES1:44</a></i>;<br/>
<a NAME="E6:51"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> is <a href="ordinal1.html#V6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a><br/>

<b>assume </b><a NAME="E1:51_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E4:51_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:51_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="card_lar.html#E4:51"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_lar.html#E5:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:51_2"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">y</font> &amp; ( <font color="Maroon">x</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x</font> ) )
 
<b>by </b><i><a class="ref" href="card_lar.html#T3" target="_self">Th3</a>, <a class="txt" href="card_lar.html#E4:51"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_lar.html#E5:51"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E9:51_2"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> or <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/>
<b>hence </b><a NAME="E10:51_2"/>
<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:51_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="card_lar.html#E4:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E5:51_3"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">x</font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_lar.html#T2" target="_self">Th2</a>, <a class="ref" href="card_lar.html#T3" target="_self">Th3</a>, <a class="txt" href="card_lar.html#E4:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_lar.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E6:51_3"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_lar.html#T3" target="_self">Th3</a>, <a class="txt" href="card_lar.html#E4:51"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:51"/><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="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T4" target="_self">Th4</a>, <a class="ref" href="card_3.html#T62">CARD_3:62</a></i>;<br/>
<a NAME="E9:51"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E10:51"/><b>then </b>
<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:51"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4">Rank</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="txt" href="card_lar.html#E4:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
