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

<b>let </b><font color="Maroon">K</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>







<b>assume </b><a NAME="E1:65"/><i><font color="Green">E44</font></i>: 
( ( <font color="Maroon">K</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">L</font> &amp; <font color="Maroon">M</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">N</font> ) or ( <font color="Maroon">K</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">L</font> &amp; <font color="Maroon">M</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">N</font> ) or ( <font color="Maroon">K</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">L</font> &amp; <font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">N</font> ) or ( <font color="Maroon">K</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">L</font> &amp; <font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">N</font> ) )
 ;<br/>

<a NAME="E2:65"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">K</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">L</font> &amp; <font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:65_1"/><b>then </b>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><a NAME="E3:65_1"/><b>then </b>
( not  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font>, <a href="xboole_0.html#K1">{}</a>  <a href="wellord2.html#R2">are_equipotent</a>  &amp; 0 <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_1.html#T46">CARD_1:46</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E4:65_1"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 0 <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><a NAME="E5:65_1"/><b>then </b>
( 0 <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span> &amp; 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> 0 )
 <b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><a NAME="E6:65_1"/><b>then </b>
(  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> 0</span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#D6">CARD_1:def 6</a></i>;<br/><a NAME="E7:65_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( 1 <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="card_1.html#T76">CARD_1:76</a></i>;<br/><a NAME="E8:65_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">M</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E9:65_1"/><b>then </b>
(  <a href="card_2.html#K3">exp</a> <font color="Maroon">K</font>,<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">K</font></span>)</span> &amp;  <a href="card_2.html#K3">exp</a> <font color="Maroon">L</font>,<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">N</font>,<font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span> &amp;  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font> )
 <b>by </b><i>, <a class="ref" href="card_2.html#D3">CARD_2:def 3</a>, <a class="ref" href="funct_5.html#T63">FUNCT_5:63</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E10:65_1"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="card_2.html#K3">exp</a> <font color="Maroon">K</font>,<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span> &amp;  <a href="card_2.html#K3">exp</a> <font color="Maroon">L</font>,<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E7:65_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_2.html#T14">CARD_2:14</a>, <a class="ref" href="card_2.html#D2">CARD_2:def 2</a>, <a class="ref" href="funct_5.html#T69">FUNCT_5:69</a></i>;<br/><a NAME="E11:65_1"/><b>then </b>
1 <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">L</font>,<font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="card_4.html#E2:65"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E12:65_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<font color="Maroon">L</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">L</font>,<font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="card_2.html#T33">CARD_2:33</a></i>;<br/><b>hence </b><a NAME="E13:65_1"/>
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> 0 or  <a href="card_2.html#K3">exp</a> <font color="Maroon">K</font>,<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">L</font>,<font color="Maroon">N</font> )
 <b>by </b><i><a class="txt" href="card_4.html#E8:65_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:65"/>
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> 0 or  <a href="card_2.html#K3">exp</a> <font color="Maroon">K</font>,<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> <font color="Maroon">L</font>,<font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T6" target="_self">Th6</a>, <a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>


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