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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E1:8"/><i><font color="Green">E10</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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="E1:8_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:8_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E4:8_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E5:8_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="stirl2_1.html#T70">STIRL2_1:70</a></i>;<br/>
<a NAME="E6:8_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:8_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="card_fin.html#E3:8_1"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:8"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><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="E1:8_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:8_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:8_2"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:8_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:8"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E4:8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="newton.html#K12">!</a>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T18">NEWTON:18</a>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/>
<a NAME="E5:8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K12">!</a> 
 
;<br/>
<b>hence </b><a NAME="E6:8"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i><a class="txt" href="card_fin.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="card_fin.html#T7" target="_self">Th7</a></i>;<br/>


</div>
