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

<b>let </b><font color="Maroon">a</font> be   <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:43"/>
 <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="card_1.html#NR3" target="_self">is_limit_cardinal</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:43"/>
<font color="Maroon">a</font> <a href="zfrefle1.html#R4">is_cofinal_with</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">xi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E5:43"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">xi</font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> )
 <b>by </b><i><a class="ref" href="zfrefle1.html#D5">ZFREFLE1:def 5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM3">T-Sequence</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">phi</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E25</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">psi</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> &amp; <font color="Olive">psi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font> holds <br/><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">psi</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S4">ORDINAL1:sch 4</a>();<br/></i>
<a NAME="E8:43"/><i><font color="Green">E26</font></i>: 
(  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>
<a NAME="E9:43"/>
<font color="Maroon">phi</font> is <a href="ordinal2.html#V1">Ordinal-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a><br/>

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

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_1"/>
<font color="Maroon">y</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E6:43_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E7:43_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:43_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">phi</font> = <font color="Maroon">phi</font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">phi</font>
;<br/>

<b>thus </b><a NAME="E11:43"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> implies <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> );<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 ;<br/><b>thus </b><a NAME="E2:43_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:43_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:43_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:43_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E5:43_2_1"/>
( <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:43_2_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T25">CARD_1:25</a></i>;<br/>
<a NAME="E7:43_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="card_1.html#D7">CARD_1:def 7</a>, <a class="ref" href="card_4.html#T22">CARD_4:22</a></i>;<br/>
<a NAME="E8:43_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E9:43_2_1"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="card_1.html#T25">CARD_1:25</a>, <a class="ref" href="card_2.html#T3">CARD_2:3</a></i>;<br/>
<a NAME="E10:43_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E11:43_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_2_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="card_5.html#T10" target="_self">Th10</a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E6:43_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T10" target="_self">Th10</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:43_2_1"/><b>then </b>
(  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="card_5.html#E8:43"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E13:43_2_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T25">CARD_1:25</a></i>;<br/>
<a NAME="E14:43_2_1"/><b>then </b>
(  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">a</font> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="card_1.html#D7">CARD_1:def 7</a>, <a class="ref" href="card_4.html#T22">CARD_4:22</a></i>;<br/>
<a NAME="E15:43_2_1"/><b>then </b>
( ( <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> or <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> ) &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#T11" target="_self">Th11</a>, <a class="ref" href="card_5.html#T12" target="_self">Th12</a>, <a class="ref" href="card_1.html#T13">CARD_1:13</a>, <a class="ref" href="ordinal3.html#T15">ORDINAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E16:43_2_1"/>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:43"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<b>thus </b><a NAME="E14:43"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_3"/>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="card_5.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E6:43_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E15:43"/>
<font color="Maroon">phi</font> is <a href="ordinal2.html#V2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:43_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> )
 ;<br/>

<a NAME="E2:43_4"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:43_4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 
<b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E5:43_4"/>
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="card_5.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>
<a NAME="E6:43_4"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T36">CARD_1:36</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E7:43_4"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:43_4"/>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E16:43"/>
<font color="Maroon">a</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">phi</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> ;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:43_5"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">x</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:43_5"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E8:43_5"/>
<font color="Maroon">y</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E9:43_5"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E10:43_5"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">A</font> &amp;  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">A</font> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T36">CARD_1:36</a></i>;<br/>
<a NAME="E11:43_5"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:43_5"/><b>then </b>
<font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:43_5"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> )
 
<b>by </b><i>, , <a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:43_5"/><b>then </b>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">phi</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E15:43_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">phi</font>
 <b>by </b><i><a class="ref" href="card_5.html#T11" target="_self">Th11</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E17:43"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_6"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_6"/>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="card_5.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E6:43_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:43"/>
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">phi</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>

<a NAME="E19:43"/>
<font color="Maroon">phi</font> is <a href="card_3.html#V1">Cardinal-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_3.html#D1">CARD_3:def 1</a><br/>

<b>assume </b><a NAME="E1:43_7"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:43_7"/>
<font color="Maroon">y</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 
<b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:43_7"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:43_7"/><b>then </b>
( <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> or <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span> ) &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal3.html#T15">ORDINAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E6:43_7"/>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> is    <a href="hidden.html#M1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:43"/>
<font color="Maroon">phi</font> is   <a href="card_3.html#NM1">Cardinal-Function</a>
 ;<br/>

<b>assume </b><a NAME="E21:43"/>
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:43"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; 0 <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E25:43"/>
<font color="Maroon">x</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font>
 
<b>by </b><i><a class="ref" href="card_5.html#T9" target="_self">Th9</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E26:43"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E27:43"/><b>then </b>
( 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p3">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="card_5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E28:43"/><b>then </b>
( 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> or 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T15">ORDINAL3:15</a></i>;<br/>
<b>hence </b><a NAME="E29:43"/>
contradiction
 <b>by </b><i><a class="ref" href="card_1.html#T33">CARD_1:33</a></i>;<br/>


</div>
