<?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="E1:49"/><i><font color="Green">E23</font></i>: 
(  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E2:49"/>
(  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <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="E3:49"/><b>then </b><i><font color="Green">E24</font></i>: 
(  <a href="card_2.html#K3">exp</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>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</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> <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>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="card_2.html#K3">exp</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>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="card_2.html#K2">*`</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="card_2.html#T43">CARD_2:43</a>, <a class="ref" href="card_4.html#T71">CARD_4:71</a></i>;<br/>
<a NAME="E4:49"/>
 <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>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon">phi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:49"/><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;  <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> &amp; <font color="Maroon">phi</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">phi</font> &amp; <font color="Maroon">phi</font> is   <a href="card_3.html#NM1">Cardinal-Function</a> &amp; not 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:49"/><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="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>,<font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; 0 <a href="hidden.html#R1">=</a> 0 &amp;  <a href="card_2.html#K3">exp</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>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</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> <span class="p2">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>,<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">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>  <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"><a href="card_3.html#K2">disjoin</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">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>  <a href="card_1.html#K1">Card</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> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span> <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><a class="ref" href="card_1.html#D5">CARD_1:def 5</a>, <a class="ref" href="card_2.html#D3">CARD_2:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</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">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">fx</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) ) ) );<br/>
<a NAME="E8:49"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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> holds <br/> ex <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </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="E1:49_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">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</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>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">fx</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) );<br/>
<a NAME="E4:49_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_1"/>
<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>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">g</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">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>;<br/>
<b>consider </b><font color="Maroon">fx</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:49_1_1"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">z</font>] implies <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>) ) &amp; (  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">z</font>] implies <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">z</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1">PARTFUN1:sch 1</a>();<br/></i>
<b>take </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
;<br/>

<b>take </b>
<font color="Maroon">fx</font>
;<br/>

<b>thus </b><a NAME="E4:49_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</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> implies <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</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> implies <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) )
 <b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:49_1"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="ref" href="card_5.html#T11" target="_self">Th11</a>);<br/></i>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E7:49_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">fx</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) ) ) )
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="card_5.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:49"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="ref" href="card_5.html#T9" target="_self">Th9</a>);<br/></i>
<a NAME="E11:49"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><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="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E50</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">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">h1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:49_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</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">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">fx</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) ) ) )
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">h2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:49_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</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">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">fx</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) ) ) )
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a>, <a class="ref" href="card_5.html#T12" target="_self">Th12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 ;<br/><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">b</font> ;<br/><a NAME="E3:49_2_1"/>
( <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>, <font color="Maroon">A2</font> = <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">A</font> = <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font>;<br/><a NAME="E5:49_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/><a NAME="E6:49_2_1"/><b>then </b>
( ( <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> or <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> ) &amp; <font color="Maroon">a</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D6">ORDINAL1:def 6</a>, <a class="ref" href="ordinal3.html#T15">ORDINAL3:15</a></i>;<br/><a NAME="E7:49_2_1"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">phi</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">phi</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/><b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:49_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> &amp;  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/><b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:49_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x2</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">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:49_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> implies <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> implies <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E15:49_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f2</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> implies <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> implies <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) )
 <b>by </b><i>, , </i>;<br/><a NAME="E16:49_2_1"/>
( <font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E17:49_2_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, , , <a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E18:49_2_1"/><b>then </b>
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E19:49_2_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:49_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</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">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:49"/><i><font color="Green">E58</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</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> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E14:49"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="E3:49_3"/><i><font color="Green">E61</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">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:49_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">fx</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> implies <font color="Olive">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) ) ) )
 <b>by </b><i><a class="ref" href="card_5.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E6:49_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</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="E1:49_3_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_3_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">fx</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:49_3_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> holds <br/>( ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> implies <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> implies <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) ) ) )
 <b>by </b><i>, <a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E6:49_3_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">fx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:49_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fx</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_3_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</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">x2</font> = <font color="Maroon">x2</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E6:49_3_1_1"/>
( ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> or  not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, , <a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:49_3_1_1"/><b>then </b>
( ( ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> ) or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 ) &amp; 0 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<b>hence </b><a NAME="E8:49_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:49_3_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">fx</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"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:49_3_1"/>
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</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="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:49_3_1"/><b>then </b>
( <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> is   <a href="card_1.html#NM1">Cardinal</a> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="card_3.html#D1">CARD_3:def 1</a></i>;<br/>
<a NAME="E10:49_3_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</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="E11:49_3_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</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"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">fx</font>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E12:49_3_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, , , , <a class="ref" href="card_3.html#T33">CARD_3:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:49_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:49"/><b>then </b><i><font color="Green">E70</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_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</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_5.html#T11" target="_self">Th11</a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<a NAME="E16:49"/>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span> <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><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then </b><i><font color="Green">E71</font></i>:  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p4">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#T68">FUNCT_5:68</a></i>
<br/>.= 
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#D3">CARD_2:def 3</a></i>
;<br/>
<a NAME="E18:49"/><i><font color="Green">E72</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</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">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="card_3.html#D3">CARD_3:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5"><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="E1:49_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:49_5"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">F</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</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_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="card_3.html#D3">CARD_3:def 3</a></i>;<br/><b>hence </b><a NAME="E3:49_5"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="card_2.html#T13">CARD_2:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:49"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="card_3.html#K1">Card</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_6"><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="E1:49_6"/><i><font color="Green">E74</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">F</font>
 ;<br/><a NAME="E2:49_6"/><b>then </b><i><font color="Green">E75</font></i>: 
( <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">F</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">F</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> )
 <b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">x</font> as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, , </i>;<br/><a NAME="E4:49_6"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, , </i>;<br/><a NAME="E5:49_6"/><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> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">M</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:49_6"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="card_2.html#D3">CARD_2:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:49"/><b>then </b>
 <a href="card_3.html#K1">Card</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="card_5.html#K2" target="_self">-powerfunc_of</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="card_5.html#T12" target="_self">Th12</a>, <a class="txt" href="card_5.html#E8:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E23:49"/><b>then </b>
(  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &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"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</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_3.html#T54">CARD_3:54</a></i>;<br/>
<a NAME="E24:49"/><b>then </b>
 <a href="card_2.html#K3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2">disjoin</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</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> <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>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T71">CARD_4:71</a></i>;<br/>
<a NAME="E25:49"/><b>then </b>
 <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_2.html#K3">exp</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>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</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/>
<b>hence </b><a NAME="E26:49"/>
 <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_2.html#K3">exp</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>,<span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
