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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">set </span><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/>
<span class="kw">set </span><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"/>
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T4" title="FUNCT_4:th.4">FUNCT_4:4</a></span>;<br/>
<a NAME="E3:43"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></span>;<br/>
<a NAME="E4:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="card_3.html#T33" title="CARD_3:th.33">CARD_3:33</a></span>;<br/>
<a NAME="E5:43"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
(  <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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:43_2"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E2:43_2"/>
 <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> 
  <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:43_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_2_1"/><span class="kw">then </span>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E3:43_2"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_2"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><a NAME="E5:43_2"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:43"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c2">a</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="card_5.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="card_5.html#T16" target="_self" title="CARD_5:th.16">Th25</a>, <a class="ref" href="card_2.html#T27" title="CARD_2:th.27">CARD_2:27</a></span>;<br/>
<a NAME="E7:43"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T93" title="CARD_2:th.93">CARD_2:93</a></span>;<br/>
<a NAME="E8:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T18" target="_self" title="CARD_5:th.18">Th27</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:43_3"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:43_3"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E4:43_3"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c6">c</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>
 ;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_3"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>;<br/><a NAME="E6:43_3"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_3"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><a NAME="E7:43_3"/>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_3"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:43_3"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_3"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="card_2.html#T93" title="CARD_2:th.93">CARD_2:93</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:43"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="card_3.html#T30" title="CARD_3:th.30">CARD_3:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:43"/>
 <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
