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

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



<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E1:47"/><i><font color="Green">E23</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_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="E1:47_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span> 
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> )
 
;<br/>
<b>hence </b><a NAME="E3:47_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_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="E1:47_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font><b> such that </b><br/><a NAME="E3:47_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> is   <a href="card_1.html#NM1">Cardinal</a> )
 ;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i/>;<br/><a NAME="E5:47_2"/>
( <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></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> <font color="Maroon">a</font>,<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> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> &amp;  <a href="card_2.html#K3">exp</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> <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>, , , <a class="ref" href="card_4.html#T71">CARD_4:71</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E6:47_2"/>
<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> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:47"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  &amp;  <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> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>

<b>thus </b><a NAME="E2:47_3"/>
 <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> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span> 
  <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="47_3_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="E1:47_3_1"/>
<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>
 ;<br/>

<a NAME="E2:47_3_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> is   <a href="card_1.html#NM1">Cardinal</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:47_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span> 
 ;<br/>


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

<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="E3:47_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span> 
 ;<br/>

<a NAME="E4:47_3"/><b>then </b>
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> )
 
;<br/>
<b>hence </b><a NAME="E5:47_3"/>
<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/>


</div><b>end;</b></div>
<a NAME="E4:47"/>
( 1 <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="card_2.html#T40">CARD_2:40</a></i>;<br/>
<a NAME="E5:47"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">a</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">a</font> <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> &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_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></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_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#T52">CARD_3:52</a>, <a class="ref" href="card_4.html#T71">CARD_4:71</a>, <a class="ref" href="funct_4.html#T5">FUNCT_4:5</a></i>;<br/>
<a NAME="E6:47"/><b>then </b>
(  <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_3.html#K6">Sum</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> &amp;  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> : <font color="Olive">c</font> is   <a href="card_1.html#NM1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <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_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, , , <a class="ref" href="card_3.html#T43">CARD_3:43</a>, <a class="ref" href="card_3.html#T46">CARD_3:46</a></i>;<br/>
<b>hence </b><a NAME="E7:47"/>
 <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><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div>
