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

<b>let </b><font color="Maroon">c<sub>1</sub></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>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E17</font></i>: 
<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="E3:18"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="card_lar.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
 <a href="card_lar.html#K1" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E5:18"/><i><font color="Green">E18</font></i>: 
 <a href="ordinal2.html#K2">inf</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#D6">ORDINAL2:def 6</a></i>;<br/>
<a NAME="E6:18"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="ordinal1.html#K3">On</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="ordinal3.html#T8">ORDINAL3:8</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green">E20</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="card_lar.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_lar.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E9:18"/>
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> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><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="E1:18_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<a NAME="E2:18_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1">CARD_FIL:sch 1</a>(<a class="txt" href="card_lar.html#E1:18_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:18"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E8:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:18"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K1" target="_self">LBound</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="card_lar.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="card_lar.html#E5:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="card_lar.html#E7:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="card_lar.html#D6" target="_self">Def6</a></i>;<br/>


</div>
