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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <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="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <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/>( <font color="Olive" title="b2">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">B1</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Olive" title="b2">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">B1</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">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="firebrick">::  thesis: </font></span></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="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <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>  holds <br/>( <font color="Olive" title="b1">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Olive" title="b1">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</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="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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c1">A</font> &amp; <font color="Maroon" title="c3">X</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Maroon" title="c3">X</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  implies ( <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">X</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  or  not <font color="Maroon" title="c3">X</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  or ( <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:20"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
( <font color="Maroon" title="c3">X</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:20"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <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="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">CARD_LAR:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c4">B</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span><br/>

<a NAME="E3:20_1"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</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="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T22" title="ORDINAL2:th.22">ORDINAL2:22</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E4:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T18" title="ORDINAL2:th.18">ORDINAL2:18</a></span>;<br/>
<a NAME="E5:20_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E6:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</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"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="ordinal2.html#T22" title="ORDINAL2:th.22">ORDINAL2:22</a></span>;<br/>
<a NAME="E7:20_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="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E8:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="card_lar.html#D5" target="_self" title="CARD_LAR:def.5">Def5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:20_1"/>
 not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E10:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E12:20_1"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E13:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E14:20_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:20_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="ordinal2.html#T18" title="ORDINAL2:th.18">ORDINAL2:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E4:20"/>
 for <font color="Olive" title="b1">B2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</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> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Olive" title="b1">B2</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="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">B2</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</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> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</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="E1:20_2"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c4">B2</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:()"><span class="comment"><font color="firebrick">::  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> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_2_1"/>
( <font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> or <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_2_1_1"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:20_2_1_1"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E4:20_2_1_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="card_lar.html#T6" target="_self" title="CARD_LAR:th.6">Th7</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font> )</span><br/><a NAME="E5:20_2_1_1"/>
<font color="Maroon" title="c2">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_1_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E4:20_2_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:20_2_1_1"/>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font></span><br/><span class="kw">thus </span><a NAME="E7:20_2_1_1"/>
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_2_1_2"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:20_2_1_2"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E4:20_2_1_2"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c2">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="card_lar.html#T6" target="_self" title="CARD_LAR:th.6">Th7</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font> )</span><br/><a NAME="E5:20_2_1_2"/>
 not <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_2_1_2"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:20_2_1_2"/>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2_1_2"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font></span><br/><span class="kw">thus </span><a NAME="E7:20_2_1_2"/>
<font color="Maroon" title="c4">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E4:20_2_1_2"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:20"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">B1</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#T6" target="_self" title="CARD_LAR:th.6">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
