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

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

<b>assume </b><a NAME="E1:72"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">M</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:72"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:72_1"/><b>then </b>
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:72_1"/>
<font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:72"/>
<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</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#K3">alef</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>;<br/>
<a NAME="E6:72"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:72"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0 )
 <b>by </b><i>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E9:72"/>
not <font color="Maroon">M</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="txt" href="card_4.html#E8:72"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="card_1.html#T83">CARD_1:83</a></i>;<br/>
<a NAME="E10:72"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>
<a NAME="E11:72"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E12:72"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="card_4.html#T7" target="_self">Th7</a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E13:72"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<a NAME="E14:72"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Olive">Z</font> is <a href="ordinal1.html#V6">c=-linear</a> holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:72_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E2:72_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<a NAME="E3:72_2"/>
(  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> is <a href="relat_1.html#V1">Relation-like</a> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> is <a href="funct_1.html#V1">Function-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">F</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>;<br/>
<b>thus </b><a NAME="E1:72_2_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> holds <br/> ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:72_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72_2_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">Y</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="card_4.html#E13:72"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E5:72_2_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:72_2_1_1"/>
 ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="card_4.html#T19" target="_self">Th19</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a><br/><b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E2:72_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_1"/><i><font color="Green">E71</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>assume </b><a NAME="E5:72_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:72_2_1"/><i><font color="Green">E73</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">X1</font>, <font color="Maroon">f2</font> = <font color="Maroon">X2</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="card_4.html#E13:72"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="card_4.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E9:72_2_1"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="card_4.html#T20" target="_self">Th20</a>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E10:72_2_1"/><b>then </b>
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> or <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E11:72_2_1"/><b>then </b>
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f1</font> holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font> ) ) or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f2</font> holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font> ) ) )
 
<b>by </b><i><a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="card_4.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:72_2_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="card_4.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E5:72_2"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:72_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:72_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_2"/><i><font color="Green">E75</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:72_2_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T20" target="_self">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:72_2_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E8:72_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T20" target="_self">Th20</a>, <a class="ref" href="card_4.html#T22" target="_self">Th22</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:72_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="card_4.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:72_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:72_2_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72_2_3"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:72_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:72_2_3"/><i><font color="Green">E110</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:72_2_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:72_2_3"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E10:72_2_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T22" target="_self">Th22</a>, <a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E11:72_2_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:72_2_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:72_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E8:72_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:72_2_4"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:72_2_4"/><b>then </b><i><font color="Green">E138</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_4"/><i><font color="Green">E139</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:72_2_4"/><i><font color="Green">E140</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:72_2_4"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T26" target="_self">Th26</a></i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:72_2_4"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E11:72_2_4"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="card_4.html#T26" target="_self">Th26</a>, , <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E12:72_2_4"/><b>then </b>
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> or <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E13:72_2_4"/><b>then </b>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> ) )
 
<b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, , <a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_4.html#T28" target="_self">Th28</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E14:72_2_4"/>
(  not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_4.html#T28" target="_self">Th28</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:72_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:72_2_5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:72_2_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:72_2_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_5_1"/><i><font color="Green">E143</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:72_2_5_1"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E7:72_2_5_1"/>
<font color="Maroon">g</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E8:72_2_5_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<a NAME="E9:72_2_5_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E10:72_2_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


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

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

<b>assume </b><a NAME="E2:72_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E3:72_2_5"/><b>then </b><i><font color="Green">E145</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:72_2_5"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:72_2_5"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:72_2_5"/><i><font color="Green">E148</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:72_2_5"/><i><font color="Green">E149</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:72_2_5"/><i><font color="Green">E150</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:72_2_5"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T28" target="_self">Th28</a></i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:72_2_5"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E17:72_2_5"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="card_4.html#T28" target="_self">Th28</a>, <a class="ref" href="card_4.html#T29" target="_self">Th29</a>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E18:72_2_5"/><b>then </b>
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> or <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E19:72_2_5"/><b>then </b>
( ( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  ) or ( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="card_4.html#T28" target="_self">Th28</a>, <a class="ref" href="card_4.html#T29" target="_self">Th29</a>, , <a class="txt" href="card_4.html#E2:72_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E20:72_2_5"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> ) or ( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> ) )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:72_2_5"/><b>then </b><i><font color="Green">E153</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, , <a class="txt" href="card_4.html#E2:72_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:72_2_5"/>
( <font color="Maroon">f1</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f2</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T28" target="_self">Th28</a>, <a class="ref" href="card_4.html#T29" target="_self">Th29</a>, , <a class="txt" href="card_4.html#E2:72_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E23:72_2_5"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<b>hence </b><a NAME="E24:72_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:72_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T20" target="_self">Th20</a>, <a class="ref" href="card_4.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:72"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> holds <br/> not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) )
 <b>by </b><i><a class="ref" href="card_4.html#T8" target="_self">Th8</a>, <a class="ref" href="orders_1.html#T177">ORDERS_1:177</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E18:72"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E19:72"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E20:72"/><b>then </b><i><font color="Green">E156</font></i>: 
 ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E21:72"/><b>then </b><i><font color="Green">E157</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_3"/><i><font color="Green">E158</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:72_3"/><i><font color="Green">E159</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_4.html#T22" target="_self">Th22</a>, , <a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/><a NAME="E4:72_3"/><b>then </b><i><font color="Green">E160</font></i>: 
(  <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; not  <a href="card_1.html#K3">alef</a> 0 is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="card_4.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E5:72_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0
 <b>by </b><i/>;<br/><a NAME="E6:72_3"/><b>then </b><i><font color="Green">E161</font></i>: 
(  <a href="card_1.html#K3">alef</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <a href="card_1.html#K3">alef</a> 0 <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:72_3"/><b>then </b><i><font color="Green">E162</font></i>: 
(  <a href="card_1.html#K3">alef</a> 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <a href="card_1.html#K3">alef</a> 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"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="txt" href="card_4.html#E3:72"><i><font color="Green">E49</font></i></a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_2.html#T47">CARD_2:47</a></i>;<br/><a NAME="E8:72_3"/>
( <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E9:72_3"/><b>then </b><i><font color="Green">E163</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E10:72_3"/><b>then </b>
( <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><a NAME="E11:72_3"/><b>then </b><i><font color="Green">E164</font></i>: 
( <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/><a NAME="E12:72_3"/><b>then </b><i><font color="Green">E165</font></i>: 
( <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><span class="p1">(<span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="card_4.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/><a NAME="E14:72_3"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then </b><i><font color="Green">E166</font></i>:  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><span class="p3"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T48">CARD_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="card_2.html#T48">CARD_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p3">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p3">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>
;<br/><a NAME="E16:72_3"/>
<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="txt" href="card_4.html#E1:72"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E17:72_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0 &amp; <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0 )
 <b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, </i>;<br/><a NAME="E18:72_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, , <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:72_3"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/><a NAME="E21:72_3"/><i><font color="Green">E168</font></i>: 
( <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="card_4.html#T28" target="_self">Th28</a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><a NAME="E22:72_3"/><b>then </b>
( <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/><a NAME="E23:72_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/><a NAME="E24:72_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><a NAME="E25:72_3"/><b>then </b><i><font color="Green">E169</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E26:72_3"/><b>then </b><i><font color="Green">E170</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a>, <a class="ref" href="funct_4.html#T19">FUNCT_4:19</a>, <a class="ref" href="funct_4.html#T33">FUNCT_4:33</a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E27:72_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/><a NAME="E28:72_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>then </b><i><font color="Green">E171</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/><i><font color="Green">E172</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , , <a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/><a NAME="E31:72_3"/><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_4.html#T7" target="_self">Th7</a>, <a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E32:72_3"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E33:72_3"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="card_4.html#T30" target="_self">Th30</a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E34:72_3"/>
<font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3_5"><b>proof </b></a><div class="add">

<a NAME="E1:72_3_5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E2:72_3_5"/><b>then </b><i><font color="Green">E175</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E3:72_3_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/>

<a NAME="E4:72_3_5"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:72_3_5"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> ) )
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>;<br/>
<b>hence </b><a NAME="E6:72_3_5"/>
(  not <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:72_3"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , , <a class="ref" href="card_4.html#T31" target="_self">Th31</a>, <a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/><a NAME="E36:72_3"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="card_4.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E37:72_3"/><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</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="card_4.html#T24" target="_self">Th24</a>, , <a class="ref" href="card_2.html#T4">CARD_2:4</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font>;<br/><a NAME="E39:72_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K3">alef</a> 0 &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T32" target="_self">Th32</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E40:72_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:72"/><b>then </b><i><font color="Green">E178</font></i>: 
( not  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T20" target="_self">Th20</a>, , <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<div><i><font color="Green">E179</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_4"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:72_4"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="txt" href="card_4.html#E8:72"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_4.html#T22" target="_self">Th22</a>, , <a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/><a NAME="E3:72_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_4.html#T20" target="_self">Th20</a>, <a class="txt" href="card_4.html#E3:72"><i><font color="Green">E49</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:72_4"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_4.html#T22" target="_self">Th22</a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/><b>set </b><font color="Maroon">B</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>;<br/><a NAME="E6:72_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/><a NAME="E7:72_4"/><b>then </b><i><font color="Green">E182</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><a NAME="E8:72_4"/>
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>;<br/><a NAME="E9:72_4"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="card_2.html#T14">CARD_2:14</a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/><a NAME="E10:72_4"/><b>then </b><i><font color="Green">E183</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_4.html#T22" target="_self">Th22</a>, , <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><a NAME="E11:72_4"/><i><font color="Green">E184</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E12:72_4"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E13:72_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_4.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E14:72_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E15:72_4"/><b>then </b><i><font color="Green">E185</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E16:72_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_2.html#T48">CARD_2:48</a></i>;<br/><b>then </b><i><font color="Green">E186</font></i>:  <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"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> = 
 <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="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="card_4.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
;<br/><a NAME="E19:72_4"/><b>then </b><i><font color="Green">E187</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E20:72_4"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></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="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="card_4.html#T27" target="_self">Th27</a>, <a class="ref" href="card_2.html#D2">CARD_2:def 2</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a>, <a class="ref" href="zfmisc_1.html#T126">ZFMISC_1:126</a></i>;<br/><a NAME="E21:72_4"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></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="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, <a class="ref" href="card_2.html#T14">CARD_2:14</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E22:72_4"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/><a NAME="E23:72_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E24:72_4"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E26:72_4"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/><a NAME="E27:72_4"/><i><font color="Green">E189</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E28:72_4"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , <a class="ref" href="funct_4.html#T33">FUNCT_4:33</a></i>;<br/><a NAME="E29:72_4"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_4.html#T19">FUNCT_4:19</a>, <a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/><a NAME="E30:72_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E31:72_4"/><b>then </b><i><font color="Green">E190</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E32:72_4"/><i><font color="Green">E191</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E33:72_4"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , , <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E34:72_4"/><b>then </b><i><font color="Green">E192</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><a NAME="E35:72_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E36:72_4"/><b>then </b><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="card_4.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E37:72_4"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E38:72_4"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>
 <b>by </b><i>, , , , <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E39:72_4"/>
<font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4_3"><b>proof </b></a><div class="add">

<a NAME="E1:72_4_3"/><i><font color="Green">E195</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E2:72_4_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E3:72_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="card_4.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:72_4_3"/><b>then </b><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E5:72_4_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/>

<a NAME="E6:72_4_3"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:72_4_3"/><b>then </b><i><font color="Green">E197</font></i>: 
( ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> ) )
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>;<br/>
<b>assume </b><a NAME="E8:72_4_3"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<b>hence </b><a NAME="E9:72_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="card_4.html#T32" target="_self">Th32</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E40:72_4"/><b>then </b><i><font color="Green">E198</font></i>: 
( <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_4.html#T13" target="_self">Th13</a>, , , , <a class="ref" href="card_4.html#T30" target="_self">Th30</a>, <a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/><a NAME="E41:72_4"/><i><font color="Green">E199</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</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="card_4.html#T22" target="_self">Th22</a>, , </i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>;<br/><a NAME="E43:72_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_4.html#T26" target="_self">Th26</a>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E44:72_4"/><b>then </b>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E45:72_4"/>
contradiction
 <b>by </b><i><a class="ref" href="card_4.html#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E200</font></i>: <font color="Maroon">M</font> <a href="card_2.html#K2">*`</a> <font color="Maroon">M</font> = 
 <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="relat_1.html#K2">rng</a> <font color="Maroon">f</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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#D2">CARD_2:def 2</a></i>
<br/>.= 
 <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="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T14">CARD_2:14</a></i>
;<br/>
<a NAME="E26:72"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E27:72"/>
<font color="Maroon">M</font> <a href="card_2.html#K2">*`</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_4.html#T24" target="_self">Th24</a>, <a class="ref" href="card_4.html#T26" target="_self">Th26</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
