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



<b>assume </b><a NAME="E1:48"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a</font> <a href="card_1.html#NR3" target="_self">is_limit_cardinal</a>  &amp; <font color="Maroon">b</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> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">L</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">A</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:48"/>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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="E1:48_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> <b>by </b><i>, <a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<a NAME="E6:48_1"/>
(  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , , <a class="ref" href="zfrefle1.html#T20">ZFREFLE1:20</a></i>;<br/>
<a NAME="E7:48_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:48_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:48"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">L</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">L</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_3.html#T54">CARD_3:54</a></i>;<br/>
<a NAME="E6:48"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:48"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#D3">CARD_2:def 3</a></i>;<br/>
<a NAME="E8:48"/><i><font color="Green">E42</font></i>: 
 <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:48"/><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E44</font></i>: 
<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="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>set </b><font color="Maroon">m</font> =  <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font>;<br/><a NAME="E3:48_2"/>
( <font color="Maroon">b</font>,<font color="Maroon">b</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">x'</font>, <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font> <a href="wellord2.html#R2">are_equipotent</a>  )
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><a NAME="E4:48_2"/><b>then </b>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">x'</font> &amp; <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">x'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x'</font> )
 <b>by </b><i>, <a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T24">CARD_1:24</a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="funct_5.html#T67">FUNCT_5:67</a></i>;<br/><a NAME="E5:48_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span>,<font color="Maroon">b</font> &amp;  <a href="card_1.html#K1">Card</a> <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>, <a class="ref" href="card_2.html#D3">CARD_2:def 3</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><a NAME="E6:48_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</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">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:48_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_5.html#T11" target="_self">Th11</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:48_2"/><b>then </b>
(  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E9:48_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_3.html#T55">CARD_3:55</a></i>;<br/><a NAME="E10:48_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E11:48_2"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="card_5.html#T11" target="_self">Th11</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:48"/><b>then </b>
(  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">L</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_3.html#K6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_3.html#K6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</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_3.html#T43">CARD_3:43</a>, <a class="ref" href="card_3.html#T52">CARD_3:52</a></i>;<br/>
<a NAME="E12:48"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">a</font> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_3.html#K6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:48"/>
<font color="Maroon">a</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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="E1:48_3"/><i><font color="Green">E47</font></i>: 
<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="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font>;<br/>
<a NAME="E3:48_3"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T24">CARD_1:24</a></i>;<br/>
<a NAME="E4:48_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font> <a href="card_1.html#NR2" target="_self">&lt;`</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="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E5:48_3"/><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> <font color="Maroon">x'</font></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> <font color="Maroon">x'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<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="E6:48_3"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<a NAME="E7:48_3"/><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> <font color="Maroon">x'</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:48_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:48_3"/><b>then </b>
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E10:48_3"/><b>then </b>
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span>
 
;<br/>
<a NAME="E11:48_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p3">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <span class="p2">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p3">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a>, <a class="ref" href="card_3.html#T55">CARD_3:55</a></i>;<br/>
<a NAME="E12:48_3"/><b>then </b>
(  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</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="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font></span>)</span></span>)</span>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:48_3"/><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> <font color="Maroon">x</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:48_3"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E15:48_3"/><b>then </b>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">x'</font> &amp;  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T36">CARD_1:36</a>, <a class="ref" href="card_4.html#T20">CARD_4:20</a></i>;<br/>
<b>hence </b><a NAME="E16:48_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:48"/><b>then </b>
<font color="Maroon">a</font> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_3.html#K6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:48"/>
 <a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</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="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
