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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E34">A1</font></i>: 
(  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c2">b</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">a</font> )
 ;<br/>

<a NAME="E2:48"/>
(  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#T45" target="_self" title="CARD_5:th.45">Th45</a></i>;<br/>
<a NAME="E3:48"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
(  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp; <font color="Maroon" title="c2">b</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E1:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="card_5.html#T27" target="_self" title="CARD_5:th.27">Th27</a>, <a class="ref" href="card_2.html#T43" title="CARD_2:th.43">CARD_2:43</a>, <a class="ref" href="card_4.html#T71" title="CARD_4:th.71">CARD_4:71</a></i>;<br/>
<a NAME="E4:48"/>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E1:48"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">phi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:48"/><i><font color="Green" title="E36">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">phi</font> &amp; <font color="Maroon" title="c3">phi</font> is   <a href="card_3.html#NM1" title="CARD_3:NM.1">Cardinal-Function</a> &amp; not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T41" target="_self" title="CARD_5:th.41">Th41</a></i>;<br/>
<a NAME="E7:48"/><i><font color="Green" title="E37">A4</font></i>: 
(  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span>)</span> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a>, <a class="ref" href="card_2.html#D3" title="CARD_2:def.3">CARD_2:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b4">fx</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b4">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b5">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> implies <font color="Olive" title="b4">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">z</font> ) &amp; ( not <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> implies <font color="Olive" title="b4">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) ) ) );<br/>
<a NAME="E8:48"/><i><font color="Green" title="E38">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">x1</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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">fx</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 implies <font color="Olive" title="b1">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> ) &amp; ( not <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 implies <font color="Olive" title="b1">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) );<br/>
<a NAME="E4:48_1"/><i><font color="Green" title="E40">A7</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">fx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green" title="E41">A8</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">z</font>] implies <font color="Maroon" title="c7">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) ) &amp; (  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">z</font>] implies <font color="Maroon" title="c7">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">z</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></i>
<b>take </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">fx</font>
;<br/>

<b>thus </b><a NAME="E4:48_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c7">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) &amp; ( not <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c7">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E3:48_1_1"><i><font color="Green" title="E41">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:48_1"/><i><font color="Green" title="E41">A9</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="card_5.html#E4:48_1"><i><font color="Green" title="E40">A7</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c6">h</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">g</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">h</font>
;<br/>

<b>thus </b><a NAME="E7:48_1"/>
( <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">fx</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> ) &amp; ( not <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E3:48_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="card_5.html#E6:48_1"><i><font color="Green" title="E41">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:48"/><i><font color="Green" title="E39">A10</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="card_5.html#E8:48"><i><font color="Green" title="E38">A5</font></i></a>);<br/></i>
<a NAME="E11:48"/><i><font color="Green" title="E40">A11</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E41">A12</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">g1</font>, <font color="Maroon" title="c8">h1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:48_2"/><i><font color="Green" title="E42">A13</font></i>: 
( <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">fx</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c8">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> ) &amp; ( not <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E10:48"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">g2</font>, <font color="Maroon" title="c10">h2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:48_2"/><i><font color="Green" title="E43">A14</font></i>: 
( <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c10">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b1">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">fx</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c10">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x2</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> ) &amp; ( not <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x2</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E10:48"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="card_5.html#E1:48_2"><i><font color="Green" title="E41">A12</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_2_1"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c11">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">b</font> ;<br/><a NAME="E3:48_2_1"/>
( <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g1</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">g2</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E3:48_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="card_5.html#E5:48_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">A1</font> = <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c14">A2</font> = <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">a</font> <b>by </b><i><a class="txt" href="card_5.html#E3:48_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="card_5.html#E5:48_2"><i><font color="Green" title="E43">A14</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c15">A</font> = <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font>;<br/><a NAME="E5:48_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="ref" href="card_5.html#T32" target="_self" title="CARD_5:th.32">Th32</a></i>;<br/><a NAME="E6:48_2_1"/><b>then </b>
( ( <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A1</font> or <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">A2</font> ) &amp; <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D6" title="ORDINAL1:def.6">ORDINAL1:def 6</a>, <a class="ref" href="ordinal3.html#T15" title="ORDINAL3:th.15">ORDINAL3:15</a></i>;<br/><a NAME="E7:48_2_1"/><b>then </b>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:48_2_1"/><i><font color="Green" title="E44">A15</font></i>: 
( <font color="Maroon" title="c16">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">B</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E6:48"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:48_2_1"/><i><font color="Green" title="E45">A16</font></i>: 
( <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">phi</font> &amp; <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E9:48_2_1"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:48_2_1"/><i><font color="Green" title="E46">A17</font></i>: 
( <font color="Maroon" title="c8">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">f1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> implies <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) &amp; ( not <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> implies <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E6:48"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="card_5.html#E3:48_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="card_5.html#E11:48_2_1"><i><font color="Green" title="E45">A16</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E15:48_2_1"/><i><font color="Green" title="E47">A18</font></i>: 
( <font color="Maroon" title="c10">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">f2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> implies <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) &amp; ( not <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> implies <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E6:48"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="card_5.html#E5:48_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="card_5.html#E11:48_2_1"><i><font color="Green" title="E45">A16</font></i></a></i>;<br/><a NAME="E16:48_2_1"/>
( <font color="Maroon" title="c13">A1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font> &amp; <font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">A2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E17:48_2_1"/><b>then </b>
( <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font> &amp; <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E1:48_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="txt" href="card_5.html#E3:48_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="card_5.html#E5:48_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="card_5.html#E9:48_2_1"><i><font color="Green" title="E44">A15</font></i></a>, <a class="txt" href="card_5.html#E13:48_2_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="card_5.html#E15:48_2_1"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E18:48_2_1"/><b>then </b>
( <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> &amp; <font color="Maroon" title="c18">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E11:48_2_1"><i><font color="Green" title="E45">A16</font></i></a>, <a class="txt" href="card_5.html#E13:48_2_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="card_5.html#E15:48_2_1"><i><font color="Green" title="E47">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E19:48_2_1"/>
<font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:48_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="card_5.html#E3:48_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="card_5.html#E5:48_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">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" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:48"/><i><font color="Green" title="E41">A19</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E14:48"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></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" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_3"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_3"/><i><font color="Green" title="E42">A20</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:48_3"/><i><font color="Green" title="E43">A21</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> holds <br/> ex <font color="Olive" title="b2">fx</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> ) &amp; ( not <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> implies <font color="Olive" title="b2">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E10:48"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="card_5.html#E3:48_3"><i><font color="Green" title="E42">A20</font></i></a></i>;<br/>
<a NAME="E6:48_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_3_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_3_1"/><i><font color="Green" title="E44">A22</font></i>: 
( <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">fx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:48_3_1"/><i><font color="Green" title="E45">A23</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">b</font> holds <br/>( ( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> implies <font color="Maroon" title="c12">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) &amp; ( not <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> implies <font color="Maroon" title="c12">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) ) )
 <b>by </b><i><a class="txt" href="card_5.html#E5:48_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="card_5.html#E3:48_3_1"><i><font color="Green" title="E44">A22</font></i></a></i>;<br/>
<a NAME="E6:48_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">fx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_3_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">fx</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">z'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_3_1_1"/><i><font color="Green" title="E46">A24</font></i>: 
( <font color="Maroon" title="c14">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">fx</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x2</font> = <font color="Maroon" title="c11">x2</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E5:48_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="card_5.html#E3:48_3_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">A</font> = <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E6:48_3_1_1"/>
( ( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</font> or  not <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</font> ) &amp; <font color="Maroon" title="c16">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="card_5.html#E6:48"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="card_5.html#E5:48_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="card_5.html#E3:48_3_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:48_3_1_1"/><b>then </b>
( ( ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z'</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</font> ) or <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E5:48_3_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="card_5.html#E3:48_3_1_1"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<b>hence </b><a NAME="E8:48_3_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:48_3_1"/><b>then </b><i><font color="Green" title="E46">A25</font></i>: 
<font color="Maroon" title="c12">fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E5:48_3_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:48_3_1"/>
<font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E6:48"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="card_5.html#E5:48_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="card_5.html#E3:48_3_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:48_3_1"/><b>then </b>
( <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> &amp; <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E6:48"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="card_5.html#E5:48_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="card_5.html#E3:48_3_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="card_3.html#D1" title="CARD_3:def.1">CARD_3:def 1</a></i>;<br/>
<a NAME="E10:48_3_1"/><b>then </b><i><font color="Green" title="E47">A26</font></i>: 
<font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_5.html#D3" target="_self" title="CARD_5:def.3">Def3</a></i>;<br/>
<a NAME="E11:48_3_1"/><b>then </b>
( <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fx</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">fx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E13:48"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E12:48_3_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_5.html#E13:48"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="card_5.html#E5:48_3_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="card_5.html#E7:48_3_1"><i><font color="Green" title="E46">A25</font></i></a>, <a class="txt" href="card_5.html#E10:48_3_1"><i><font color="Green" title="E47">A26</font></i></a>, <a class="ref" href="card_3.html#T33" title="CARD_3:th.33">CARD_3:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:48_3"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_5.html#E3:48_3"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="card_5.html#E5:48_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:48"/><b>then </b><i><font color="Green" title="E42">A27</font></i>: 
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E7:48"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="card_5.html#E10:48"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="card_5.html#E11:48"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="card_1.html#T26" title="CARD_1:th.26">CARD_1:26</a></i>;<br/>
<a NAME="E16:48"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>then </b><i><font color="Green" title="E43">A28</font></i>:  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p3">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#T68" title="FUNCT_5:th.68">FUNCT_5:68</a></i>
<br/>.= 
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#D3" title="CARD_2:def.3">CARD_2:def 3</a></i>
;<br/>
<a NAME="E18:48"/><i><font color="Green" title="E44">A29</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a>, <a class="ref" href="card_3.html#D3" title="CARD_3:def.3">CARD_3:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_5"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font>
 ;<br/><a NAME="E2:48_5"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_5.html#E18:48"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a>, <a class="ref" href="card_3.html#D3" title="CARD_3:def.3">CARD_3:def 3</a></i>;<br/><b>hence </b><a NAME="E3:48_5"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="card_2.html#T13" title="CARD_2:th.13">CARD_2:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:48"/><b>then </b><i><font color="Green" title="E45">A30</font></i>: 
 <a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E18:48"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_6"/><i><font color="Green" title="E46">A31</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font>
 ;<br/><a NAME="E2:48_6"/><b>then </b><i><font color="Green" title="E47">A32</font></i>: 
( <span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E13:48"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c6">x</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_5.html#E13:48"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="card_5.html#E1:48_6"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="card_5.html#D3" target="_self" title="CARD_5:def.3">Def3</a></i>;<br/><a NAME="E4:48_6"/>
<font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="card_5.html#E13:48"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="card_5.html#E1:48_6"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="card_5.html#D3" target="_self" title="CARD_5:def.3">Def3</a></i>;<br/><a NAME="E5:48_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="card_5.html#D3" target="_self" title="CARD_5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E6:48_6"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="card_5.html#E2:48_6"><i><font color="Green" title="E47">A32</font></i></a>, <a class="ref" href="card_2.html#D3" title="CARD_2:def.3">CARD_2:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:48"/><b>then </b>
 <a href="card_3.html#K1" title="CARD_3:func.1">Card</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E13:48"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="card_5.html#E18:48"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E23:48"/><b>then </b>
(  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_5.html#E20:48"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="card_3.html#T54" title="CARD_3:th.54">CARD_3:54</a></i>;<br/>
<a NAME="E24:48"/><b>then </b>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T71" title="CARD_4:th.71">CARD_4:71</a></i>;<br/>
<a NAME="E25:48"/><b>then </b>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_5.html#E15:48"><i><font color="Green" title="E42">A27</font></i></a>, <a class="txt" href="card_5.html#E17:48"><i><font color="Green" title="E43">A28</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E26:48"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <span class="p1">(<span class="default"><a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_5.html#E3:48"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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