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

<b>let </b><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">a</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span><br/>



<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c4">f</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span>;<br/>
<a NAME="E1:43"/><i><font color="Green" title="E31">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E2:43_1"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> )
 
;<br/>
<b>hence </b><a NAME="E3:43_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E32">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><b>then consider </b><font color="Maroon" title="c6">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E3:43_2"/><i><font color="Green" title="E34">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">c</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="#E3:43_2"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/><a NAME="E5:43_2"/>
( <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <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>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <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>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c1">b</font> &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c1">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font> )
 <b>by </b><i><a class="txt" href="#E1:43_2"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="#E3:43_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="card_5.html#D3" target="_self" title="CARD_5:def.3">Def3</a>, <a class="ref" href="card_3.html#T139" title="CARD_3:th.139">CARD_3:139</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E6:43_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:43"/><i><font color="Green" title="E33">A5</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> </span><br/>

<b>thus </b><a NAME="E2:43_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1"><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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:43_3_1"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> </span><br/>

<a NAME="E2:43_3_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">x</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> )
 
<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="E3:43_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  or <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E3:43_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/>

<a NAME="E4:43_3"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> )
 
;<br/>
<b>hence </b><a NAME="E5:43_3"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:43"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">b</font> &amp; <font color="Maroon" title="c2">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_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="card_5.html#T25" target="_self" title="CARD_5:th.25">Th25</a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a></i>;<br/>
<a NAME="E5:43"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font> &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">a</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="card_3.html#T52" title="CARD_3:th.52">CARD_3:52</a>, <a class="ref" href="card_3.html#T139" title="CARD_3:th.139">CARD_3:139</a>, <a class="ref" href="funct_4.html#T5" title="FUNCT_4:th.5">FUNCT_4:5</a></i>;<br/>
<a NAME="E6:43"/><b>then </b>
(  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> &amp;  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">a</font> : <font color="Olive" title="b1">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> </span>}</span>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</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="c2">a</font>,<font color="Maroon" title="c1">b</font> )
 
<b>by </b><i><a class="txt" href="#E2:43"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="#E3:43"><i><font color="Green" title="E33">A5</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_3.html#T43" title="CARD_3:th.43">CARD_3:43</a>, <a class="ref" href="card_3.html#T46" title="CARD_3:th.46">CARD_3:46</a></i>;<br/>
<b>hence </b><a NAME="E7:43"/>
 <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="card_5.html#K2" title="CARD_5:func.2">-powerfunc_of</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
