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

<a NAME="E1:62_1"/>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>,<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T53" target="_self">Th53</a>, <a class="ref" href="card_2.html#T15">CARD_2:15</a></i>;<br/>
<a NAME="E2:62_1"/><b>then </b>
(  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K13">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K13">:]</a></span> )
 
<b>by </b><i><a class="ref" href="card_4.html#T53" target="_self">Th53</a>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a>, <a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:62_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>2</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>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K13">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K13">:]</a></span> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>);<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:62_1"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &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>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E7:62_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_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> ; <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="E1:62_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:62_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>] )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="numbers.html#K5">NAT</a>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:62_1_1"/><i><font color="Green">E47</font></i>: 
( <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>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="card_4.html#E4:62_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:62_1_1"/>
( <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a>, <a class="ref" href="mcart_1.html#T50">MCART_1:50</a></i>;<br/>
<a NAME="E9:62_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E4:62_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_4.html#E6:62_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_4.html#E3:62_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="card_4.html#E7:62_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:62_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E4:62_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_4.html#E6:62_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_4.html#E7:62_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:62_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span>  is <a href="card_4.html#V1" target="_self">countable</a>
 <b>by </b><i><a class="txt" href="card_4.html#E6:62_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="card_4.html#T45" target="_self">Th45</a></i>;<br/>


</div>
