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

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



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
( <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/>
<font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:4_1_1"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#T50">CARD_1:50</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a>, , <a class="ref" href="altcat_1.html#T3">ALTCAT_1:3</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><b>hence </b><a NAME="E3:4_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</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">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">X</font>, <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st not <font color="Olive">Y</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Olive">X</font>,<font color="Olive">Y</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="Olive">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Olive">X</font></span>)</span>;<br/><a NAME="E2:4_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:4_1_2_1"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:4_1_2_1"/>
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E3:4_1_2_1"/><b>then </b>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T64">FUNCT_5:64</a></i>;<br/>
<a NAME="E4:4_1_2_1"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T50">CARD_1:50</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E5:4_1_2_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</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">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:4_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

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



<b>assume </b><b>that </b><br/><a NAME="E2:4_1_2_2"/><i><font color="Green">E59</font></i>: 
not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E3:4_1_2_2"/><i><font color="Green">E60</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E4:4_1_2_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:4_1_2_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font>,<font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E8:4_1_2_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:4_1_2_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a>, , <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>1 <a href="hidden.html#R1">=</a> 1;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:4_1_2_2"/><i><font color="Green">E64</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E13:4_1_2_2"/><i><font color="Green">E68</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E14:4_1_2_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) </span>}</span> 
 <b>from </b><i><a class="ref" href="stirl2_1.html#S8">STIRL2_1:sch 8</a>(, , );<br/></i>
<b>set </b><font color="Maroon">F1</font> = <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] </span>}</span> ;<br/>
<a NAME="E15:4_1_2_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</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="E1:4_1_2_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 ;<br/>

<a NAME="E2:4_1_2_2_1"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1_2_2_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:4_1_2_2"/>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</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="E1:4_1_2_2_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:4_1_2_2_2"/>
<font color="Maroon">G</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1_2_2_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:4_1_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">cY</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="newton.html#T99">NEWTON:99</a></i>;<br/>
<a NAME="E19:4_1_2_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cY</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_1_2_2_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">Xx</font> = <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">F2</font> = <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> : <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">F3</font> = <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) </span>}</span> ;<br/>
<a NAME="E2:4_1_2_2_3"/><i><font color="Green">E79</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> : <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</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="E1:4_1_2_2_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> : <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> </span>}</span> 
 ;<br/>

<a NAME="E2:4_1_2_2_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:4_1_2_2_3_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:4_1_2_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> : <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</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="E1:4_1_2_2_3_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:4_1_2_2_3_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:4_1_2_2_3_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> : <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:4_1_2_2_3"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> : <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>] &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E5:4_1_2_2_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:4_1_2_2_3"/><b>then </b>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:4_1_2_2_3"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">Y</font> : <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> </span>}</span>  &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cY</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:4_1_2_2_3"/>
<font color="Maroon">F</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cY</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:4_1_2_2"/><b>then </b>
( ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">cY</font> ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">cY</font> ) )
 
;<br/>
<a NAME="E21:4_1_2_2"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T50">STIRL2_1:50</a>, <a class="ref" href="stirl2_1.html#T51">STIRL2_1:51</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E22:4_1_2_2"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E23:4_1_2_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</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">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:4_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>hence </b><a NAME="E5:4_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">Y</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">Y</font></span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
