<?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>assume </b><a NAME="E1:37"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0 <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>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:37"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">S</font> <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>and </b><br/><a NAME="E3:37"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Olive" title="b1">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Olive" title="b1">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )
 ;<br/>

<b>thus </b><a NAME="E4:37"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">S</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>
 <b>by </b><i><a class="txt" href="card_lar.html#E3:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="card_lar.html#T22" target="_self" title="CARD_LAR:th.22">Th22</a></i>;<br/>

<a NAME="E5:37"/>
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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/> ex <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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="E1:37_1"/><i><font color="Green" title="E31">A4</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/>

<b>reconsider </b><font color="Maroon" title="c4">B11</font> = <font color="Maroon" title="c3">B1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_lar.html#E1:37_1"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> $1,<font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>
<a NAME="E3:37_1"/><i><font color="Green" title="E32">A5</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">X</font></span>)</span> where <font color="Olive" title="b2">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>set </b><font color="Maroon" title="c6">ChB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span> ;<br/>
<a NAME="E1:37_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">X</font> being    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:37_1_1_1"/><i><font color="Green" title="E34">A7</font></i>: 
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <b>and </b><br/><a NAME="E4:37_1_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:37_1_1_1"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<a NAME="E5:37_1_1_1"/>
<font color="Maroon" title="c8">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:37"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/>
<a NAME="E6:37_1_1_1"/><b>then </b>
not <font color="Maroon" title="c8">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="card_lar.html#T8" target="_self" title="CARD_LAR:th.8">Th8</a></i>;<br/>
<a NAME="E7:37_1_1_1"/><b>then </b>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 
;<br/>
<b>hence </b><a NAME="E8:37_1_1_1"/>
<font color="Maroon" title="c7">x</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#E3:37_1_1_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:37_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:37_1"/><i><font color="Green" title="E33">A8</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> (  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of $1 =  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,$1;<br/>
<a NAME="E1:37_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</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="c2">S</font>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2" title="TREES_2:sch.2">TREES_2:sch 2</a>();<br/></i>
<a NAME="E2:37_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</font>) <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="c2">S</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:37_1"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<a NAME="E3:37_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</font>) <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#E2:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1_2"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</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="ref" href="card_5.html#T38" title="CARD_5:th.38">CARD_5:38</a></i>;<br/>

<b>consider </b><font color="Maroon" title="c6">X</font> being    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>
<a NAME="E6:37_1_2"/><i><font color="Green" title="E34">A9</font></i>: 
( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:37"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/>
<a NAME="E7:37_1_2"/><b>then </b>
not <font color="Maroon" title="c6">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="card_lar.html#T8" target="_self" title="CARD_LAR:th.8">Th8</a></i>;<br/>
<a NAME="E8:37_1_2"/><b>then </b>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">LB</font> =  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</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="c1">M</font> ;<br/>
<a NAME="E10:37_1_2"/>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c5">B</font>,<font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span> 
 
;<br/>
<a NAME="E11:37_1_2"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
( <font color="Maroon" title="c7">LB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</font>) )
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="card_lar.html#T3" target="_self" title="CARD_LAR:th.3">Th3</a></i>;<br/>
<a NAME="E12:37_1_2"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LB</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E6:37_1_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="card_lar.html#T10" target="_self" title="CARD_LAR:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E13:37_1_2"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">B</font>)
 <b>by </b><i><a class="txt" href="card_lar.html#E11:37_1_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></font>($2) &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $3 );<br/>
<a NAME="E5:37_1"/><i><font color="Green" title="E34">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>  ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">y</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></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="c1">M</font> <b>by </b><i><a class="txt" href="card_lar.html#E4:37_1"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">y</font>
;<br/>

<b>thus </b><a NAME="E2:37_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>]
 <b>by </b><i><a class="txt" href="card_lar.html#E4:37_1"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E7:37_1"/><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B11</font>
 <b>and </b><br/><a NAME="E8:37_1"/><i><font color="Green" title="E36">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="card_lar.html#E5:37_1"><i><font color="Green" title="E34">A11</font></i></a>);<br/></i>
<b>take </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L</font></span>)</span>
;<br/>

<b>reconsider </b><font color="Maroon" title="c6">L1</font> = <font color="Maroon" title="c5">L</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E10:37_1"/><i><font color="Green" title="E37">A14</font></i>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">L1</font></span>)</span> <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#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="card_lar.html#T23" target="_self" title="CARD_LAR:th.23">Th23</a></i>;<br/>
<a NAME="E11:37_1"/><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c3">B1</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="c5">L</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:37_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E12:37_1"/><b>then </b><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">RNG</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E14:37_1"/>
for <font color="Olive" title="b1">C1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RNG</font> holds <br/> ex <font color="Olive" title="b2">C2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RNG</font> &amp; <font color="Olive" title="b1">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">C1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:37_1_4"/><i><font color="Green" title="E40">A17</font></i>: 
<font color="Maroon" title="c8">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RNG</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_1_4"/><i><font color="Green" title="E41">A18</font></i>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">L</font>
 <b>and </b><br/><a NAME="E4:37_1_4"/><i><font color="Green" title="E42">A19</font></i>: 
<font color="Maroon" title="c8">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y1</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:37_1_4"><i><font color="Green" title="E40">A17</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="c10">y</font> = <font color="Maroon" title="c9">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="card_lar.html#E3:37_1_4"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">L1</font> = <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">L1</font>
;<br/>

<b>thus </b><a NAME="E7:37_1_4"/>
<font color="Maroon" title="c11">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RNG</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>

<b>thus </b><a NAME="E8:37_1_4"/>
<font color="Maroon" title="c8">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">L1</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E8:37_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="card_lar.html#E4:37_1_4"><i><font color="Green" title="E42">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">SUPL</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c7">RNG</font> as  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_lar.html#E10:37_1"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="card_lar.html#E11:37_1"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="card_lar.html#T4" target="_self" title="CARD_LAR:th.4">Th4</a></i>;<br/>
<a NAME="E16:37_1"/>
for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c8">SUPL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_1_5"/><i><font color="Green" title="E40">A20</font></i>: 
<font color="Maroon" title="c9">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c9">X1</font> as    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="card_lar.html#E1:37_1_5"><i><font color="Green" title="E40">A20</font></i></a></i>;<br/>
<a NAME="E3:37_1_5"/><i><font color="Green" title="E41">A21</font></i>: 
( <font color="Maroon" title="c10">X</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c10">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> &amp; <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:37"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/>
<a NAME="E4:37_1_5"/><b>then </b><i><font color="Green" title="E42">A22</font></i>: 
not <font color="Maroon" title="c10">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="card_lar.html#T8" target="_self" title="CARD_LAR:th.8">Th8</a></i>;<br/>
<a NAME="E5:37_1_5"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">SUPL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_5_1"/>
<font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E2:37_1_5_1"/><b>then </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T30" title="ORDINAL2:th.30">ORDINAL2:30</a></i>;<br/>
<a NAME="E3:37_1_5_1"/><b>then </b><i><font color="Green" title="E43">A23</font></i>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></i>;<br/>
<b>assume </b><a NAME="E4:37_1_5_1"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">SUPL</font>
 ;<br/>

<a NAME="E5:37_1_5_1"/><b>then </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:37_1_5_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E6:37_1_5_1"/><b>then </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">B3</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:37_1_5_1"/><i><font color="Green" title="E44">A24</font></i>: 
<font color="Maroon" title="c11">B3</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="c5">L</font>
 <b>and </b><br/><a NAME="E9:37_1_5_1"/><i><font color="Green" title="E45">A25</font></i>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">B3</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:37_1_5_1"/><i><font color="Green" title="E46">A26</font></i>: 
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">L</font>
 <b>and </b><br/><a NAME="E12:37_1_5_1"/><i><font color="Green" title="E47">A27</font></i>: 
<font color="Maroon" title="c11">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y1</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E8:37_1_5_1"><i><font color="Green" title="E44">A24</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="c13">y</font> = <font color="Maroon" title="c12">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="card_lar.html#E11:37_1_5_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:37_1_5_1"/>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:37_1_5"><i><font color="Green" title="E42">A22</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">LBY</font> =  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c10">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E16:37_1_5_1"/>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is    <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> </span>}</span> 
 
;<br/>
<a NAME="E17:37_1_5_1"/><b>then </b>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>)
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:37_1"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<a NAME="E18:37_1_5_1"/><b>then </b>
<font color="Maroon" title="c14">LBY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>)
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E19:37_1_5_1"/><b>then </b><i><font color="Green" title="E48">A28</font></i>: 
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="card_lar.html#E8:37_1"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/>
<a NAME="E20:37_1_5_1"/>
<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c5">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E21:37_1_5_1"/><b>then </b>
<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E22:37_1_5_1"/><b>then </b>
<font color="Maroon" title="c14">LBY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E19:37_1_5_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E23:37_1_5_1"/><b>then </b>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:37_1_5"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E24:37_1_5_1"/><b>then </b>
<font color="Maroon" title="c14">LBY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">SUPL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E25:37_1_5_1"/><b>then </b>
<font color="Maroon" title="c14">LBY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E9:37_1_5_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="card_lar.html#E12:37_1_5_1"><i><font color="Green" title="E47">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E26:37_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E3:37_1_5"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="card_lar.html#T10" target="_self" title="CARD_LAR:th.10">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_1_5"/>
<font color="Maroon" title="c8">SUPL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X1</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E3:37_1_5"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:37_1"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E18:37_1"/>
<font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_lar.html#E12:37_1"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:37"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">S</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 <b>by </b><i><a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</a></i>;<br/>


</div>
