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

<span class="kw">let </span><font color="Maroon" title="c1">M</font> be  non  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> implies for <font color="Olive" title="b1">S</font> being 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>  st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b2">X</font> being   <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Olive" title="b1">S</font> holds <br/> ( <font color="Olive" title="b2">X</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Olive" title="b2">X</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  ) ) holds <br/>(  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">S</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp;  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">S</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being 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>  st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b2">X</font> being   <a href="card_lar.html#M1" title="CARD_LAR:mode.1">Element</a> of <font color="Olive" title="b1">S</font> holds <br/> ( <font color="Olive" title="b2">X</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Olive" title="b2">X</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  ) ) holds <br/>(  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">S</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp;  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">S</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  )</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> &amp; ( 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>  ) ) implies (  <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>  &amp;  <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>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E3:38"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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>  &amp;  <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>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:38"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="card_lar.html#T22" target="_self" title="CARD_LAR:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<a NAME="E5:38"/>
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="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies  ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">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="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">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="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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> ;<br/>
<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>   <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/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>( $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="E3:38_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
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="38_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<span class="kw">set </span><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="E1:38_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="38_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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>  or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1_1_1"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:38_1_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E4:38_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>
 ;<br/>
<a NAME="E5:38_1_1_1"/>
<font color="Maroon" title="c8">X</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/>
<a NAME="E6:38_1_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c8">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#T8" target="_self" title="CARD_LAR:th.8">Th8</a></span>;<br/>
<a NAME="E7:38_1_1_1"/><span class="kw">then </span>
 <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/>
<span class="kw">hence </span><a NAME="E8:38_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:38_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:38_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
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="38_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> &amp; <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>) )</span><br/>

<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>   <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="E1:38_1_2"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#T3" target="_self" title="CARD_LAR:th.3">Th3</a></span>;<br/>
<a NAME="E2:38_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="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="trees_2.html#S2" title="TREES_2:sch.2">TREES_2:sch 2</a>();<br/></span>
<a NAME="E3:38_1_2"/><span class="kw">then </span>
 <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="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/>
<a NAME="E4:38_1_2"/><span class="kw">then </span>
 <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:38_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T38" title="CARD_5:th.38">CARD_5:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">consider </span><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="E7:38_1_2"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c6">X</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/>
<a NAME="E8:38_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c6">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#T8" target="_self" title="CARD_LAR:th.8">Th8</a></span>;<br/>
<a NAME="E9:38_1_2"/><span class="kw">then </span>
 <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/>
<span class="kw">then </span><span class="kw">reconsider </span><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="E11:38_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="E12:38_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/>
<a NAME="E13:38_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38_1_2"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="card_lar.html#T10" target="_self" title="CARD_LAR:th.10">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:38_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38_1_2"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E1:38_1_2"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:38_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
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="38_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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="b2">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="Maroon" title="c5">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c5">n</font>,<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>]</span><br/>

<span class="kw">thus </span><a NAME="E2:38_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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:38_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B11</font>
 <span class="kw">and </span><br/><a NAME="E8:38_1"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E5:38_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>);<br/></span>
<span class="kw">reconsider </span><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> ,<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/>
<span class="kw">take </span>
 <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> &amp; <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> )</span><br/>

<a NAME="E10:38_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E12:38_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
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="38_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">C1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies  ex <font color="Olive" title="b1">C2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RNG</font> &amp; <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1_4"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RNG</font> &amp; <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:38_1_4"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">and </span><br/><a NAME="E4:38_1_4"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">L1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> )</span><br/>

<span class="kw">thus </span><a NAME="E7:38_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">thus </span><a NAME="E8:38_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E4:38_1_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:38_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="c6">L1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="card_lar.html#T23" target="_self" title="CARD_LAR:th.23">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E12:38_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="card_lar.html#T4" target="_self" title="CARD_LAR:th.4">Th4</a></span>;<br/>
<a NAME="E15:38_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="38_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1_5"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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> ;<br/>
<a NAME="E3:38_1_5"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/>
<a NAME="E4:38_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A18</font></span>: 
not <font color="Maroon" title="c10">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#T8" target="_self" title="CARD_LAR:th.8">Th8</a></span>;<br/>
<a NAME="E5:38_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="38_1_5_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:38_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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T30" title="ORDINAL2:th.30">ORDINAL2:30</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E2:38_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A19</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:38_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:38_1_5_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_5_1"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">B3</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E6:38_1_5_1"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:38_1_5_1"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:38_1_5_1"/><span class="lab"><font color="Green" title="E43">A22</font></span>: 
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">and </span><br/><a NAME="E10:38_1_5_1"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:38_1_5_1"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38_1_5_1"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:38_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_5"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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="E14:38_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="E15:38_1_5_1"/><span class="kw">then </span>
 <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/>
<a NAME="E16:38_1_5_1"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<a NAME="E17:38_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A24</font></span>: 
 <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a></span>;<br/>
<a NAME="E18:38_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#K2" title="NAT_1:func.2">+</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<a NAME="E19:38_1_5_1"/><span class="kw">then </span>
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">SUPL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<a NAME="E20:38_1_5_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:38_1_5_1"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/>
<a NAME="E21:38_1_5_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_5"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E22:38_1_5_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<a NAME="E23:38_1_5_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38_1_5_1"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E10:38_1_5_1"><span class="lab"><font color="Green" title="E44">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:38_1_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_5"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="card_lar.html#T10" target="_self" title="CARD_LAR:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:38_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_5"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:38_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E17:38_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:38_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:38"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
