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<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>let </b><font color="Maroon">S</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">X</font> being   <a href="card_lar.html#M1" target="_self">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">X</font> is <a href="card_lar.html#V2" target="_self">closed</a>
 ;<br/>

<b>let </b><font color="Maroon">C</font> be  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_lar.html#D5" target="_self">CARD_LAR:def 5</a><br/>

<b>assume </b><a NAME="E2:35"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:35_1_1"/>
not  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:35_1_1_1"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:35_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:35_1_1"/>
(  not  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> or <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:35_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E3:35_1_2"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </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="E1:35_1_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as    <a href="card_lar.html#M1" target="_self">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:35_1_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X1</font> is <a href="card_lar.html#V2" target="_self">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:35_1_2_1"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2_1_1"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E2:35_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E3:35_1_2_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>
<a NAME="E4:35_1_2_1_1"/>
<font color="Maroon">X1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:35_1_2_1_1"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>
<a NAME="E6:35_1_2_1_1"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>
<b>hence </b><a NAME="E7:35_1_2_1_1"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:35_1_2_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:35_1_2"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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