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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>assume </b><a NAME="E1:72"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">c<sub>2</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> );<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:72_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:72_1"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:72"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0 )
 <b>by </b><i><a class="ref" href="card_4.html#T54" target="_self">Th54</a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E9:72"/>
not <font color="Maroon">c<sub>2</sub></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#E1:72"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_4.html#T7" target="_self">Th7</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">E55</font></i>: 
 <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E8:72"><i><font color="Green">E54</font></i></a>, <a class="txt" href="card_4.html#E10:72"><i><font color="Green">E55</font></i></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">E56</font></i>: 
<font color="Maroon">c<sub>3</sub></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#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E8:72"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E14:72"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="ordinal1.html#V6">c=-linear</a> holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></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">E57</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:72_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font> is <a href="relat_1.html#V1">Relation-like</a> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>thus </b><a NAME="E1:72_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></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">E59</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="card_4.html#E4:72"><i><font color="Green">E53</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E3:72_2_1_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E5:72_2_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E3:72_2_1_1"><i><font color="Green">E59</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_1"/><i><font color="Green">E59</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:72_2_1"/><i><font color="Green">E60</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="card_4.html#E4:72"><i><font color="Green">E53</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E4:72_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_4.html#E7:72_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E9:72_2_1"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E2:72_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E4:72_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_4.html#E7:72_2_1"><i><font color="Green">E60</font></i></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">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) or ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E4:72_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_4.html#E7:72_2_1"><i><font color="Green">E60</font></i></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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E4:72_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="card_4.html#E7:72_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E5:72_2"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></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">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_2"/><i><font color="Green">E60</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></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">c<sub>8</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E4:72_2_2"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E4:72_2_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="card_4.html#E7:72_2_2"><i><font color="Green">E61</font></i></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="card_4.html#E7:72_2_2"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:72_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72_2_3"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E3:72_2_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:72_2_3"/><i><font color="Green">E61</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></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">c<sub>9</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E6:72_2_3"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></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">E62</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E6:72_2_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="card_4.html#E9:72_2_3"><i><font color="Green">E62</font></i></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E9:72_2_3"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:72_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E5:72_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E8:72_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:72_2_4"/><b>then </b><i><font color="Green">E62</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:72_2_4"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:72_2_4"/><i><font color="Green">E64</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E2:72_2_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></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">E65</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E4:72_2_4"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></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">E66</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></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">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E6:72_2_4"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E11:72_2_4"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E2:72_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E4:72_2_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="card_4.html#E6:72_2_4"><i><font color="Green">E64</font></i></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">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E4:72_2_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="card_4.html#E6:72_2_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="card_4.html#E8:72_2_4"><i><font color="Green">E65</font></i></a>, <a class="txt" href="card_4.html#E10:72_2_4"><i><font color="Green">E66</font></i></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">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E8:72_2_4"><i><font color="Green">E65</font></i></a>, <a class="txt" href="card_4.html#E10:72_2_4"><i><font color="Green">E66</font></i></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></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">E62</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></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">E63</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E4:72_2_5_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E7:72_2_5_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E4:72_2_5_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_4.html#E6:72_2_5_1"><i><font color="Green">E63</font></i></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">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E4:72_2_5_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_4.html#E6:72_2_5_1"><i><font color="Green">E63</font></i></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">c<sub>7</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">E62</font></i>: 
( <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></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">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:72_2_5"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:72_2_5"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72_2_5"><i><font color="Green">E62</font></i></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">E65</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E5:72_2_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="card_4.html#E7:72_2_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:72_2_5"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:72_2_5"/><i><font color="Green">E67</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E8:72_2_5"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></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">E68</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></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">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E10:72_2_5"><i><font color="Green">E66</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></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">E69</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></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">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E1:72_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E12:72_2_5"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E17:72_2_5"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E2:72_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E10:72_2_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="card_4.html#E12:72_2_5"><i><font color="Green">E67</font></i></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">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></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">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E10:72_2_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="card_4.html#E12:72_2_5"><i><font color="Green">E67</font></i></a>, <a class="txt" href="card_4.html#E14:72_2_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="card_4.html#E16:72_2_5"><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">c<sub>7</sub></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">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>13</sub></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">E70</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E3:72_2_5"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_4.html#E14:72_2_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="card_4.html#E16:72_2_5"><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">c<sub>12</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E10:72_2_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="card_4.html#E12:72_2_5"><i><font color="Green">E67</font></i></a>, <a class="txt" href="card_4.html#E14:72_2_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="card_4.html#E16:72_2_5"><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">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E21:72_2_5"><i><font color="Green">E70</font></i></a></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E7:72_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="card_4.html#E8:72_2"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:72"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> not <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="card_4.html#E13:72"><i><font color="Green">E56</font></i></a>, <a class="ref" href="orders_1.html#T177">ORDERS_1:177</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E18:72"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E16:72"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E19:72"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E16:72"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E20:72"/><b>then </b><i><font color="Green">E59</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></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">c<sub>6</sub></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">E61</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">E62</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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_4.html#E1:72_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="card_4.html#T2" target="_self">Th2</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">E63</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">c<sub>6</sub></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">c<sub>6</sub></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#T11" target="_self">Th11</a>, <a class="ref" href="card_4.html#T41" target="_self">Th41</a>, <a class="ref" href="card_4.html#T72" target="_self">Th72</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><a class="ref" href="card_4.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E6:72_3"/><b>then </b><i><font color="Green">E64</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">c<sub>6</sub></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">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E4:72_3"><i><font color="Green">E63</font></i></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">E65</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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="card_4.html#T76" target="_self">Th76</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">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">E66</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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">E67</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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">E68</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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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="txt" href="card_4.html#E11:72_3"><i><font color="Green">E67</font></i></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">E69</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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E12:72_3"><i><font color="Green">E68</font></i></a>, <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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E7:72_3"><i><font color="Green">E65</font></i></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E7:72_3"><i><font color="Green">E65</font></i></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">c<sub>6</sub></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">c<sub>6</sub></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="txt" href="card_4.html#E7:72_3"><i><font color="Green">E65</font></i></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">c<sub>6</sub></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">c<sub>6</sub></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#T54" target="_self">Th54</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="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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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="txt" href="card_4.html#E3:72_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="card_4.html#T67" target="_self">Th67</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">c<sub>6</sub></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">c<sub>6</sub></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="txt" href="card_4.html#E4:72_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_4.html#T34" target="_self">Th34</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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="card_4.html#E7:72_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="card_4.html#E15:72_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:72_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></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">c<sub>9</sub></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">E71</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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>9</sub></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="txt" href="card_4.html#E9:72_3"><i><font color="Green">E66</font></i></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></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="txt" href="card_4.html#E21:72_3"><i><font color="Green">E71</font></i></a>, <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">c<sub>10</sub></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">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E20:72_3"><i><font color="Green">E70</font></i></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">E72</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></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">E73</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>10</sub></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">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>10</sub></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">c<sub>6</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>10</sub></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">c<sub>6</sub></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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_4.html#E26:72_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>then </b><i><font color="Green">E74</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E26:72_3"><i><font color="Green">E73</font></i></a>, <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">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="card_4.html#E20:72_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/><i><font color="Green">E75</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E20:72_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_4.html#E26:72_3"><i><font color="Green">E73</font></i></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">E76</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">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E10:72"><i><font color="Green">E55</font></i></a>, <a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E20:72"><i><font color="Green">E59</font></i></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></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">E77</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E29:72_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_4.html#E30:72_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_4.html#E31:72_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E34:72_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E20:72_3"><i><font color="Green">E70</font></i></a>, <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">E78</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>10</sub></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">c<sub>11</sub></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">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E3:72_3_5"/>
( <font color="Maroon">c<sub>11</sub></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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 ;<br/>

<a NAME="E4:72_3_5"/><b>then </b>
( ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> ) &amp; ( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E26:72_3"><i><font color="Green">E73</font></i></a>, <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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; ( <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E20:72_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_4.html#E25:72_3"><i><font color="Green">E72</font></i></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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E2:72_3_5"><i><font color="Green">E78</font></i></a>, <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">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E29:72_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_4.html#E30:72_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_4.html#E33:72_3"><i><font color="Green">E77</font></i></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">E78</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E16:72"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E37:72_3"/><i><font color="Green">E79</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">c<sub>9</sub></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#E3:72_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_4.html#E6:72_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_2.html#T4">CARD_2:4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></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">c<sub>9</sub></font>;<br/><a NAME="E39:72_3"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K3">alef</a> 0 &amp; not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E37:72_3"><i><font color="Green">E79</font></i></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="txt" href="card_4.html#E29:72_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_4.html#E36:72_3"><i><font color="Green">E78</font></i></a>, <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">E61</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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E21:72"><i><font color="Green">E60</font></i></a>, <a class="ref" href="card_4.html#T1" target="_self">Th1</a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<div><i><font color="Green">E62</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">c<sub>6</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E1:72"><i><font color="Green">E51</font></i></a>, <a class="txt" href="card_4.html#E23:72"><i><font color="Green">E61</font></i></a>, <a class="ref" href="card_4.html#T34" target="_self">Th34</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">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E21:72"><i><font color="Green">E60</font></i></a>, <a class="ref" href="card_4.html#T76" target="_self">Th76</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:72_4"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_4.html#E23:72"><i><font color="Green">E61</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E6:72_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="card_4.html#E5:72_4"><i><font color="Green">E63</font></i></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">E64</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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></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">E65</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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_4.html#E23:72"><i><font color="Green">E61</font></i></a>, <a class="ref" href="card_4.html#T33" target="_self">Th33</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><a NAME="E11:72_4"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_4.html#E5:72_4"><i><font color="Green">E63</font></i></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="card_4.html#E11:72_4"><i><font color="Green">E66</font></i></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">E67</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_4.html#E7:72_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="card_4.html#E10:72_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="card_2.html#T48">CARD_2:48</a></i>;<br/><b>then </b><i><font color="Green">E68</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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="card_4.html#E10:72_4"><i><font color="Green">E65</font></i></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="card_4.html#E15:72_4"><i><font color="Green">E67</font></i></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">E69</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E10:72_4"><i><font color="Green">E65</font></i></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">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E7:72_4"><i><font color="Green">E64</font></i></a>, <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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_4.html#E17:72_4"><i><font color="Green">E68</font></i></a>, <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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="card_4.html#E7:72_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E26:72_4"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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">E71</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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E26:72_4"><i><font color="Green">E70</font></i></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">c<sub>6</sub></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></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">c<sub>6</sub></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></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">c<sub>6</sub></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">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></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">c<sub>6</sub></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">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></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">c<sub>6</sub></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E19:72_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="card_4.html#E26:72_4"><i><font color="Green">E70</font></i></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">E74</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E5:72_4"><i><font color="Green">E63</font></i></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">E75</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E20:72"><i><font color="Green">E59</font></i></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></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">E76</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E26:72_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_4.html#E31:72_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_4.html#E34:72_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_4.html#E36:72_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E39:72_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></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">E77</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_4.html#E5:72_4"><i><font color="Green">E63</font></i></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">c<sub>6</sub></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">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E26:72_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_4.html#E1:72_4_3"><i><font color="Green">E77</font></i></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">E78</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>11</sub></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">c<sub>12</sub></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">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E5:72_4_3"/>
( <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 ;<br/>

<a NAME="E6:72_4_3"/><b>then </b>
( ( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> ) &amp; ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E32:72_4"><i><font color="Green">E73</font></i></a>, <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">E79</font></i>: 
( ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> implies <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; ( <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> implies <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E26:72_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_4.html#E27:72_4"><i><font color="Green">E71</font></i></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>hence </b><a NAME="E9:72_4_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E4:72_4_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="card_4.html#E7:72_4_3"><i><font color="Green">E79</font></i></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">E77</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E3:72"><i><font color="Green">E52</font></i></a>, <a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_4.html#E26:72_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_4.html#E31:72_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_4.html#E34:72_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_4.html#E38:72_4"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/><a NAME="E41:72_4"/><i><font color="Green">E78</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></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#E23:72"><i><font color="Green">E61</font></i></a>, <a class="txt" href="card_4.html#E7:72_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_4.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E43:72_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_4.html#E5:72_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="card_4.html#E41:72_4"><i><font color="Green">E78</font></i></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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_4.html#E26:72_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_4.html#E31:72_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_4.html#E41:72_4"><i><font color="Green">E78</font></i></a>, <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="txt" href="card_4.html#E16:72"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_4.html#E40:72_4"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E18:72"><i><font color="Green">E58</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E27:72"/>
<font color="Maroon">c<sub>2</sub></font> <a href="card_2.html#K2">*`</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E24:72"><i><font color="Green">E62</font></i></a>, <a class="txt" href="card_4.html#E25:72"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
