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

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( not $1 is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Olive" title="b1">A</font> );<br/>
<a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E2:5"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">K</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">K</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Olive" title="b1">K</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">K</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">K</font>]
 <b>and </b><br/><a NAME="E2:5_1"/><i><font color="Green" title="E9">A4</font></i>: 
not  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c2">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1"/>
<font color="Maroon" title="c2">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c3">K'</font> = <font color="Maroon" title="c2">K</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E3:5_1_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">K'</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/><a NAME="E4:5_1_1"/><b>then </b>
 <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">K'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a></i>;<br/><b>hence </b><a NAME="E5:5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_5.html#E2:5_1"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:5_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="card_5.html#E1:5_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>take </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font>
;<br/>

<b>thus </b><a NAME="E6:5_1"/>
 <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_5.html#E5:5_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="card_1.html#T39" title="CARD_1:th.39">CARD_1:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:5"/><i><font color="Green" title="E8">A6</font></i>: 
for <font color="Olive" title="b1">K</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">K</font> <a href="card_1.html#NR3" title="CARD_1:NR.3">is_limit_cardinal</a>  &amp; ( for <font color="Olive" title="b2">N</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Olive" title="b2">N</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">N</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">K</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">K</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_2"/><i><font color="Green" title="E9">A7</font></i>: 
( <font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">K</font> <a href="card_1.html#NR3" title="CARD_1:NR.3">is_limit_cardinal</a>  )
 <b>and </b><br/><a NAME="E2:5_2"/><i><font color="Green" title="E10">A8</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>  st <font color="Olive" title="b1">N</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">N</font>]
 <b>and </b><br/><a NAME="E3:5_2"/><i><font color="Green" title="E11">A9</font></i>: 
not <font color="Maroon" title="c2">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">N</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> st <br/>( <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; not <font color="Olive" title="b1">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> );<br/>
<b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:5_2"/><i><font color="Green" title="E12">A10</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Olive" title="b1">A</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> );<br/>
<a NAME="E6:5_2"/><i><font color="Green" title="E13">A11</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T42" title="CARD_1:th.42">CARD_1:42</a></i>;<br/>
<a NAME="E7:5_2"/><i><font color="Green" title="E14">A12</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:5_2_1"/><i><font color="Green" title="E15">A13</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">N</font> being   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a><b> such that </b><br/><a NAME="E3:5_2_1"/><i><font color="Green" title="E16">A14</font></i>: 
( <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; not <font color="Maroon" title="c5">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/>
<a NAME="E4:5_2_1"/>
<font color="Maroon" title="c5">N</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="card_5.html#E1:5_2_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="txt" href="card_5.html#E3:5_2_1"><i><font color="Green" title="E16">A14</font></i></a></i>;<br/>
<a NAME="E5:5_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Olive" title="b1">A</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E2:5_2"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="card_5.html#E3:5_2_1"><i><font color="Green" title="E16">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:5_2_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:5_2"/><i><font color="Green" title="E15">A15</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="card_5.html#E6:5_2"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="card_5.html#E7:5_2"><i><font color="Green" title="E14">A12</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2"><b>now </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> ;<br/><b>assume </b><a NAME="E1:5_2_2"/>
<font color="Maroon" title="c5">x</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="c4">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:5_2_2"/><i><font color="Green" title="E16">A16</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:5_2_2"/><i><font color="Green" title="E17">A17</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="card_5.html#E3:5_2_2"><i><font color="Green" title="E16">A16</font></i></a></i>;<br/><b>thus </b><a NAME="E6:5_2_2"/>
<font color="Maroon" title="c5">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>by </b><i><a class="txt" href="card_5.html#E5:5_2_2"><i><font color="Green" title="E17">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E7:5_2_2"/>
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z</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="E1:5_2_2_1"/><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">z'</font> = <font color="Maroon" title="c8">z</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E5:5_2_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:5_2_2_1"/>
(  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">z'</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c7">A</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="card_5.html#E3:5_2_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="card_5.html#E5:5_2_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="card_5.html#E1:5_2_2_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="card_1.html#T41" title="CARD_1:th.41">CARD_1:41</a></i>;<br/>
<a NAME="E4:5_2_2_1"/><b>then </b>
(  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; not  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">z'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="ref" href="card_5.html#T11" target="_self" title="CARD_5:th.11">Th11</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E5:5_2_2_1"/><b>then </b><i><font color="Green" title="E19">A19</font></i>: 
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/>
<a NAME="E6:5_2_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>(  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">z'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a></i>;<br/>
<a NAME="E7:5_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">z'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T42" title="CARD_1:th.42">CARD_1:42</a></i>;<br/>
<b>hence </b><a NAME="E8:5_2_2_1"/>
<font color="Maroon" title="c8">z</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="c4">f</font>
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="card_5.html#E5:5_2_2_1"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">A</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T31" title="ORDINAL1:th.31">ORDINAL1:31</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">A</font>
;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c6">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E13:5_2"/><i><font color="Green" title="E16">A20</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <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="c5">A</font> holds <br/><font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:5_2_3"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:5_2_3"/><i><font color="Green" title="E17">A21</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:5_2_3"/><i><font color="Green" title="E18">A22</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="card_5.html#E3:5_2_3"><i><font color="Green" title="E17">A21</font></i></a></i>;<br/><a NAME="E6:5_2_3"/>
(  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">C</font></span>)</span> &amp;  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c9">C</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="card_5.html#E3:5_2_3"><i><font color="Green" title="E17">A21</font></i></a>, <a class="txt" href="card_5.html#E5:5_2_3"><i><font color="Green" title="E18">A22</font></i></a>, <a class="ref" href="card_1.html#T39" title="CARD_1:th.39">CARD_1:39</a></i>;<br/><a NAME="E7:5_2_3"/><b>then </b>
(  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">K</font> &amp;  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E1:5_2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a>, <a class="ref" href="card_4.html#T22" title="CARD_4:th.22">CARD_4:22</a></i>;<br/><a NAME="E8:5_2_3"/><b>then </b>
(  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; not  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="card_5.html#T11" target="_self" title="CARD_5:th.11">Th11</a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/><a NAME="E9:5_2_3"/><b>then </b><i><font color="Green" title="E19">A23</font></i>: 
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E11:5_2_3"/><i><font color="Green" title="E20">A24</font></i>: 
(  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c10">D</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">D</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a></i>;<br/><a NAME="E12:5_2_3"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">D</font>
 <b>by </b><i><a class="txt" href="card_5.html#E11:5_2_3"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="card_1.html#T42" title="CARD_1:th.42">CARD_1:42</a></i>;<br/><b>hence </b><a NAME="E13:5_2_3"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="card_5.html#E5:5_2_3"><i><font color="Green" title="E18">A22</font></i></a>, <a class="txt" href="card_5.html#E9:5_2_3"><i><font color="Green" title="E19">A23</font></i></a>, <a class="txt" href="card_5.html#E11:5_2_3"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:5_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<a NAME="E16:5_2"/><b>then </b><i><font color="Green" title="E17">A25</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E13:5_2"><i><font color="Green" title="E16">A20</font></i></a>, <a class="ref" href="card_1.html#T40" title="CARD_1:th.40">CARD_1:40</a></i>;<br/>
<a NAME="E17:5_2"/>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_2_4"/><i><font color="Green" title="E18">A26</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c7">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:5_2_4"/><i><font color="Green" title="E19">A27</font></i>: 
( <font color="Maroon" title="c9">C</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="c6">L</font> &amp; <font color="Maroon" title="c8">x'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">C</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E1:5_2_4"><i><font color="Green" title="E18">A26</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:5_2_4"/><i><font color="Green" title="E20">A28</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E4:5_2_4"><i><font color="Green" title="E19">A27</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E6:5_2_4"><i><font color="Green" title="E20">A28</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:5_2_4"/><i><font color="Green" title="E21">A29</font></i>: 
( <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c11">y</font> &amp; not  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c11">y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="card_5.html#E13:5_2"><i><font color="Green" title="E16">A20</font></i></a>, <a class="txt" href="card_5.html#E6:5_2_4"><i><font color="Green" title="E20">A28</font></i></a>, <a class="ref" href="card_5.html#T11" target="_self" title="CARD_5:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:5_2_4"/><i><font color="Green" title="E22">A30</font></i>: 
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E13:5_2"><i><font color="Green" title="E16">A20</font></i></a>, <a class="txt" href="card_5.html#E6:5_2_4"><i><font color="Green" title="E20">A28</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:5_2_4"/>
 ex <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="card_5.html#E10:5_2_4"><i><font color="Green" title="E22">A30</font></i></a></i>;<br/>
<a NAME="E12:5_2_4"/><b>then </b>
( <font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="card_5.html#E8:5_2_4"><i><font color="Green" title="E21">A29</font></i></a>, <a class="txt" href="card_5.html#E10:5_2_4"><i><font color="Green" title="E22">A30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:5_2_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="card_5.html#E4:5_2_4"><i><font color="Green" title="E19">A27</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:5_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E19:5_2"/><b>then </b><i><font color="Green" title="E18">A31</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_2_5_1"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_5_1_1"><b>case </b></a><a NAME="E1:5_2_5_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_2_5_1_1"/><b>then </b>
( not  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; not  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="ref" href="card_5.html#T11" target="_self" title="CARD_5:th.11">Th11</a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><a NAME="E3:5_2_5_1_1"/><b>then </b>
not  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/><a NAME="E4:5_2_5_1_1"/><b>then </b>
( <font color="Maroon" title="c2">K</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E3:5_2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a>, <a class="ref" href="card_4.html#T11" title="CARD_4:th.11">CARD_4:11</a></i>;<br/><b>hence </b><a NAME="E5:5_2_5_1_1"/>
<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_5_1_2"><b>case </b></a><a NAME="E1:5_2_5_1_2"/><i><font color="Green" title="E19">A32</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:5_2_5_1_2"/>
<font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c5">A</font>
 ;<br/><a NAME="E3:5_2_5_1_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; not  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c5">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="card_5.html#E16:5_2"><i><font color="Green" title="E17">A25</font></i></a>, <a class="txt" href="card_5.html#E19:5_2"><i><font color="Green" title="E18">A31</font></i></a>, <a class="txt" href="card_5.html#E1:5_2_5_1_2"><i><font color="Green" title="E19">A32</font></i></a>, <a class="ref" href="card_5.html#T11" target="_self" title="CARD_5:th.11">Th11</a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/><a NAME="E4:5_2_5_1_2"/><b>then </b><i><font color="Green" title="E20">A33</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="card_5.html#E5:5_2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="card_5.html#E16:5_2"><i><font color="Green" title="E17">A25</font></i></a>, <a class="txt" href="card_5.html#E1:5_2_5_1_2"><i><font color="Green" title="E19">A32</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:5_2_5_1_2"/><i><font color="Green" title="E21">A34</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c6">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a></i>;<br/><a NAME="E7:5_2_5_1_2"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="card_5.html#E16:5_2"><i><font color="Green" title="E17">A25</font></i></a>, <a class="txt" href="card_5.html#E1:5_2_5_1_2"><i><font color="Green" title="E19">A32</font></i></a>, <a class="txt" href="card_5.html#E6:5_2_5_1_2"><i><font color="Green" title="E21">A34</font></i></a>, <a class="ref" href="card_1.html#T42" title="CARD_1:th.42">CARD_1:42</a></i>;<br/><a NAME="E8:5_2_5_1_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="card_5.html#E9:5_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="card_5.html#E4:5_2_5_1_2"><i><font color="Green" title="E20">A33</font></i></a>, <a class="txt" href="card_5.html#E6:5_2_5_1_2"><i><font color="Green" title="E21">A34</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:5_2_5_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E21:5_2"/>
<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Maroon" title="c5">A</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:5"/>
for <font color="Olive" title="b1">M</font> being  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">M</font>]
 
<b>from </b><i><a class="ref" href="card_1.html#S1" title="CARD_1:sch.1">CARD_1:sch 1</a>(<a class="txt" href="card_5.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="card_5.html#E2:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="card_5.html#E3:5"><i><font color="Green" title="E8">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:5"/>
( not <font color="Maroon" title="c1">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <font color="Olive" title="b1">A</font> )
 ;<br/>


</div>
