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

<b>let </b><font color="Maroon">Fy</font> be  <a href="finset_1.html#V2">finite-yielding</a> <a href="funct_1.html#NM1">Function</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="E1:84"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">XF</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:84"/><i><font color="Green">E50</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XF</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green">E51</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="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XF</font> holds <br/> ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; ( for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Olive">x</font>,<font color="Olive">y</font> holds <br/> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Olive">f</font>,<font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">XF</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">n</font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>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="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">XF</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">n</font></span>)</span></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">X</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E4:84"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:84_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T8">STIRL2_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">x'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">XF</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">x'</font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">ch</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">ch</font>
;<br/>

<b>thus </b><a NAME="E6:84_1"/>
( <font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">ch</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">ch</font>] )
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>, <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E6:84"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E7:84"/><i><font color="Green">E59</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E10:84"/>
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="Olive">n</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="card_fin.html#K7" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_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="E1:84_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:84_2"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; ( for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Olive">x</font>,<font color="Olive">y</font> holds <br/> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Olive">f</font>,<font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">XF</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font> ) )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E3:84_2"/><b>then </b>
(  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">XF</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">F</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">XF</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></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">X</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="card_fin.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:84_2"/>
<font color="Maroon">F</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:84"/>
(  <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">X</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">F</font> &amp; ( 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="Olive">n</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="card_fin.html#K7" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">XF</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">n</font></span>)</span></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">X</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, , , </i>;<br/>


</div>
