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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 ;<br/>

<b>assume </b><a NAME="E2:32"/>
not  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">B1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E5:32"/><i><font color="Green" title="E27">A3</font></i>: 
 <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font>
 <b>by </b><i><a class="ref" href="card_lar.html#T5" target="_self" title="CARD_LAR:th.5">Th5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">B1</font>
;<br/>

<b>set </b><font color="Maroon" title="c4">SUCC</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font> ) </span>}</span> ;<br/>
<a NAME="E6:32"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font> ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">B2</font>
 <b>and </b><br/><a NAME="E4:32_1"/>
( <font color="Maroon" title="c6">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">B2</font> )
 <b>by </b><i><a class="txt" href="card_lar.html#E1:32_1"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:32_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E3:32_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">SUCC</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font> ) </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E8:32"/>
for <font color="Olive" title="b1">B</font> being <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">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"><font color="Maroon" title="c5">SUCC</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SUCC</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">B</font> be  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">B0</font> = <font color="Maroon" title="c6">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="card_lar.html#E1:32_2"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/>
<a NAME="E3:32_2"/>
not  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">SUCC</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1"/><i><font color="Green" title="E29">A7</font></i>: 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">SUCC</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 ;<br/>

<a NAME="E2:32_2_1"/>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1_1"/>
not  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">B5</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:32_2_1_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c8">B5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E4:32_2_1_1"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B5</font>
 <b>by </b><i><a class="ref" href="card_lar.html#T6" target="_self" title="CARD_LAR:th.6">Th6</a></i>;<br/>
<a NAME="E5:32_2_1_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">B5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:32_2_1_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<a NAME="E6:32_2_1_1"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">B5</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">B6</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E8:32_2_1_1"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c9">B6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SUCC</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E9:32_2_1_1"/><i><font color="Green" title="E33">A11</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">B5</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">B6</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:32_2_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<a NAME="E10:32_2_1_1"/>
<font color="Maroon" title="c9">B6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SUCC</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E8:32_2_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">B7</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E12:32_2_1_1"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c9">B6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">B7</font>
 <b>and </b><br/><a NAME="E13:32_2_1_1"/><i><font color="Green" title="E35">A13</font></i>: 
( <font color="Maroon" title="c10">B7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">B7</font> )
 ;<br/>
<a NAME="E14:32_2_1_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">B5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">B7</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E9:32_2_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="card_lar.html#E12:32_2_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E15:32_2_1_1"/><b>then </b><i><font color="Green" title="E36">A14</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">B5</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">B7</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E16:32_2_1_1"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c9">B6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E8:32_2_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E17:32_2_1_1"/>
<font color="Maroon" title="c10">B7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">B7</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<a NAME="E18:32_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">B7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E12:32_2_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="card_lar.html#E16:32_2_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E19:32_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">B7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E13:32_2_1_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E20:32_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">B7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B5</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:32_2_1_1"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:32_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E15:32_2_1_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_2_1"/><b>then </b>
<font color="Maroon" title="c7">B0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B10</font> where <font color="Olive" title="b1">B10</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( not <font color="Olive" title="b1">B10</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">B10</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">B10</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B10</font> ) </span>}</span> 
 
;<br/>
<a NAME="E4:32_2_1"/><b>then </b><i><font color="Green" title="E30">A16</font></i>: 
<font color="Maroon" title="c7">B0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c2">X</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c8">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">B3</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:32_2_1"/><i><font color="Green" title="E31">A17</font></i>: 
<font color="Maroon" title="c9">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SUCC</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E8:32_2_1"/>
<font color="Maroon" title="c8">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">B3</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:32_2_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<a NAME="E9:32_2_1"/><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c9">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:32_2_1"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:32_2_1"/>
<font color="Maroon" title="c9">B3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SUCC</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:32_2_1"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">B4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E12:32_2_1"/><i><font color="Green" title="E33">A19</font></i>: 
<font color="Maroon" title="c9">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">B4</font>
 <b>and </b><br/><a NAME="E13:32_2_1"/>
<font color="Maroon" title="c10">B4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E14:32_2_1"/><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">B4</font>
 ;<br/>
<b>thus </b><a NAME="E15:32_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E5:32"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="card_lar.html#E4:32_2_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="card_lar.html#E9:32_2_1"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="card_lar.html#E12:32_2_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="card_lar.html#E14:32_2_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_2"/>
(  not  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">SUCC</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> or <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SUCC</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:32"/><b>then </b><i><font color="Green" title="E28">A21</font></i>: 
<font color="Maroon" title="c5">SUCC</font> <a href="card_lar.html#R2" title="CARD_LAR:pred.2">is_closed_in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D2" target="_self" title="CARD_LAR:def.2">Def2</a></i>;<br/>
<a NAME="E10:32"/>
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="c5">SUCC</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="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">B2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:32_3"/><i><font color="Green" title="E29">A22</font></i>: 
<font color="Maroon" title="c6">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">B21</font> = <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B1</font>;<br/>
<a NAME="E2:32_3"/><i><font color="Green" title="E30">A23</font></i>: 
( <font color="Maroon" title="c6">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:32_3"/>
<font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:32"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_lar.html#E1:32_3"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="ordinal3.html#T15" title="ORDINAL3:th.15">ORDINAL3:15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:32_3"/><i><font color="Green" title="E31">A24</font></i>: 
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E6:32_3"/><i><font color="Green" title="E32">A25</font></i>: 
<font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">D</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</a></i>;<br/>
<b>take </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">D</font>
;<br/>

<a NAME="E7:32_3"/>
<font color="Maroon" title="c6">B2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">D</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E6:32_3"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E8:32_3"/><b>then </b>
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">D</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:32_3"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E9:32_3"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SUCC</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E5:32_3"><i><font color="Green" title="E31">A24</font></i></a></i>;<br/>

<a NAME="E10:32_3"/>
<font color="Maroon" title="c6">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">D</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:32_3"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="card_lar.html#E6:32_3"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:32_3"/><b>then </b>
<font color="Maroon" title="c6">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">D</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<b>hence </b><a NAME="E12:32_3"/>
<font color="Maroon" title="c6">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">D</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32"/><b>then </b>
<font color="Maroon" title="c5">SUCC</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 
<b>by </b><i><a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</a></i>;<br/>
<a NAME="E12:32"/><b>then </b>
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c5">SUCC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self" title="CARD_LAR:def.4">Def4</a></i>;<br/>
<a NAME="E13:32"/><b>then </b>
<font color="Maroon" title="c5">SUCC</font> <a href="card_lar.html#R1" title="CARD_LAR:pred.1">is_unbounded_in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self" title="CARD_LAR:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E14:32"/>
( <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font> ) </span>}</span>  <a href="card_lar.html#R3" title="CARD_LAR:pred.3">is_club_in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="card_lar.html#E4:32"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_lar.html#E9:32"><i><font color="Green" title="E28">A21</font></i></a>, <a class="ref" href="card_lar.html#D3" target="_self" title="CARD_LAR:def.3">Def3</a></i>;<br/>


</div>
