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

<b>let </b><font color="Maroon" title="c1">A</font> be  <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="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> &amp; <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Olive" title="b1">B1</font>,<font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Olive" title="b1">B1</font>,<font color="Olive" title="b2">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> &amp; <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Olive" title="b1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> &amp; <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies (  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font> ) )</span><br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or (  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font> ) )</span><br/>

<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font> )</span><br/>

<a NAME="E3:18"/>
not <font color="Maroon" title="c3">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="card_lar.html#T8" target="_self" title="CARD_LAR:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
 <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 );<br/>
<b>set </b><font color="Maroon" title="c4">LB</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span> ;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#D6" title="ORDINAL2:def.6">ORDINAL2:def 6</a></i>;<br/>
<a NAME="E6:18"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E7:18"/><b>then </b><i><font color="Green" title="E19">A4</font></i>: 
 <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green" title="E20">A5</font></i>: 
 ex <font color="Olive" title="b1">B3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Olive" title="b1">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">B2</font> where <font color="Olive" title="b2">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B2</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="card_lar.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="card_lar.html#T9" target="_self" title="CARD_LAR:th.9">Th9</a></i>;<br/>
<a NAME="E9:18"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">B2</font> where <font color="Olive" title="b2">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B2</font>] </span>}</span>  holds <br/><font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span>  implies <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E2:18_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_lar.html#E1:18_1"><i><font color="Green" title="E21">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:18"/><b>then </b>
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E8:18"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:18"/>
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K1" title="CARD_LAR:func.1">LBound</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="card_lar.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="card_lar.html#E5:18"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="card_lar.html#E7:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="card_lar.html#D6" target="_self" title="CARD_LAR:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
