<?xml version="1.0"?>
<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">Fy</font> being <a href="finset_1.html#V2">finite-yielding</a> <a href="funct_1.html#NM1">Function</a><br/> for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st  <a href="relat_1.html#K1">dom</a> <font color="Olive">Fy</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &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/>for <font color="Olive">XFS</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>   st  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">XFS</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">X</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="Olive">XFS</font> holds <br/><font color="Olive">XFS</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="Olive">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> ) holds <br/> <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="Olive">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Olive">XFS</font>;<br/>
<a NAME="E1:82"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><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="E1:82_1"/><i><font color="Green">E50</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> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">XFS</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:82_1"/><i><font color="Green">E51</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</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="E3:82_1"/>
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">XFS</font> holds <br/><font color="Maroon">XFS</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>
 ;<br/>

<a NAME="E4:82_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, , <a class="ref" href="card_2.html#T59">CARD_2:59</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:82_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K44">addint</a>  )
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a>, <a class="ref" href="stirl2_1.html#D3">STIRL2_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:82_1"/>
 <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">XFS</font>
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="binop_2.html#T4">BINOP_2:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:82"/><i><font color="Green">E53</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="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><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="E1:82_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<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="E2:82_2"/><i><font color="Green">E58</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> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon">XFS</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:82_2"/><i><font color="Green">E59</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</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="E4:82_2"/><i><font color="Green">E60</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">XFS</font> holds <br/><font color="Maroon">XFS</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>
 ;<br/>

<a NAME="E5:82_2"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font> is <a href="finset_1.html#V1">finite</a> &amp; ( 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="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font> holds <br/><font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:82_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>

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

<b>assume </b><a NAME="E2:82_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font>
 ;<br/>

<a NAME="E3:82_2_1"/><b>then </b>
 ex <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> &amp; <font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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="E4:82_2_1"/>
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">urngFy</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_2_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_1"><b>suppose </b></a><a NAME="E1:82_2_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:82_2_2_1"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/><a NAME="E3:82_2_2_1"/><b>then </b>
( <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K6">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> 0</span>)</span></span><a href="afinsq_1.html#K6">%&gt;</a></span> &amp; 0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; <font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> 0 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E4:82_2_2_1"/><b>then </b>
( <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> 0 &amp; <font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> 0 <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> 0</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">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> 0 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="stirl2_1.html#T44">STIRL2_1:44</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E5:82_2_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,1
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:82_2_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</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>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E8:82_2_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T14">FUNCT_1:14</a></i>;<br/><a NAME="E9:82_2_2_1"/><b>then </b>
( <font color="Maroon">Fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T15">FUNCOP_1:15</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/><b>hence </b><a NAME="E10:82_2_2_1"/>
 <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">XFS</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_2"><b>suppose </b></a><a NAME="E1:82_2_2_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:82_2_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, <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>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">FyX</font> = <font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</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>;<br/><b>set </b><font color="Maroon">I</font> =  <a href="yellow20.html#K1">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</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></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>;<br/><a NAME="E4:82_2_2_2"/>
<font color="Maroon">urngFy</font> <a href="hidden.html#R1">=</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> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , </i>;<br/><a NAME="E5:82_2_2_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">urngFy</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">urngFyX</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E7:82_2_2_2"/>
<font color="Maroon">urngFyX</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">urngI</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E9:82_2_2_2"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</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></span>)</span> <a href="hidden.html#R1">=</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>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>consider </b><font color="Maroon">XFyX</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E11:82_2_2_2"/><i><font color="Green">E71</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFyX</font> <a href="hidden.html#R1">=</a>  <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>
 <b>and </b><br/><a NAME="E12:82_2_2_2"/><i><font color="Green">E73</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">XFyX</font> holds <br/><font color="Maroon">XFyX</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> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<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/>;<br/><b>consider </b><font color="Maroon">XI</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E14:82_2_2_2"/><i><font color="Green">E74</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XI</font> <a href="hidden.html#R1">=</a>  <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>
 <b>and </b><br/><a NAME="E15:82_2_2_2"/><i><font color="Green">E75</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">XI</font> holds <br/><font color="Maroon">XI</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> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<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/>;<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>  = <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">XI</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E16:82_2_2_2"/><i><font color="Green">E77</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> <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> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">XI1</font> being   <a href="funct_2.html#NM1">Function</a> of  <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="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E18:82_2_2_2"/><i><font color="Green">E79</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> <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> holds <br/><font color="Maroon">XI1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i><a NAME="E19:82_2_2_2"/><i><font color="Green">E80</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">XI1</font> <a href="hidden.html#R1">=</a>  <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> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">XI1</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">XI1</font> = <font color="Maroon">XI1</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">XI1</font> = <font color="Maroon">XI1</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="txt" href="card_fin.html#E19:82_2_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/><b>set </b><font color="Maroon">Fyx</font> = <font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>;<br/><a NAME="E22:82_2_2_2"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  &amp; 0 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E23:82_2_2_2"/><b>then </b><i><font color="Green">E81</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K4">INT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K4">INT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">XI1</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">XFyX</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T36">AFINSQ_1:36</a>, <a class="ref" href="subset_1.html#T55">SUBSET_1:55</a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">XcF</font> = <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span>, <font color="Maroon">X0</font> = <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> 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/><a NAME="E25:82_2_2_2"/>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></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">XI1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">XFyX</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="txt" href="card_fin.html#E23:82_2_2_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E26:82_2_2_2"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">XI1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">XFyX</font> <a href="ordinal4.html#K1">^</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T29">AFINSQ_1:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">F1</font> = <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">XI1</font>, <font color="Maroon">F2</font> = <font color="Maroon">XFyX</font> <a href="ordinal4.html#K1">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> 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/><a NAME="E28:82_2_2_2"/>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XI1</font> <a href="hidden.html#R1">=</a>  <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> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFyX</font> <a href="hidden.html#R1">=</a>  <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> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &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">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_fin.html#E19:82_2_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#T37">AFINSQ_1:37</a>, <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a></i>;<br/><a NAME="E29:82_2_2_2"/><b>then </b><i><font color="Green">E82</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/><a NAME="E30:82_2_2_2"/>
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">XFS</font> holds <br/><font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2_2_2_1"><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="E1:82_2_2_2_1"/><i><font color="Green">E86</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">XFS</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_2_2_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:82_2_2_2_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:82_2_2_2_1_1_1"/>
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFyX</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_fin.html#E11:82_2_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_1"/><b>then </b>
( <font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">XFyX</font> <a href="card_fin.html#K7" target="_self">.</a> 0 &amp; <font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> 0 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">XFyX</font> <a href="card_fin.html#K7" target="_self">.</a> 0 <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> 0</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> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E15:82_2_2_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="afinsq_1.html#T39">AFINSQ_1:39</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E4:82_2_2_2_1_1_1"/><b>then </b>
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> 0</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>;<br/><a NAME="E5:82_2_2_2_1_1_1"/><b>then </b>
( <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> 0</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,1 &amp; <font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> 0 <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> 0</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">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> 0 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="txt" href="card_fin.html#E2:82"><i><font color="Green">E53</font></i></a>, <a class="txt" href="card_fin.html#E9:82_2_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="card_fin.html#T5" target="_self">Th5</a>, , <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E6:82_2_2_2_1_1_1"/>
<font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:82_2_2_2_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1 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="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_2"/>
(  <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"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">XFS</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> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="card_fin.html#E9:82_2_2_2"><i><font color="Green">E70</font></i></a>, , <a class="txt" href="card_fin.html#E12:82_2_2_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="card_fin.html#T5" target="_self">Th5</a>, , <a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E4:82_2_2_2_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">XFS</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="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p3">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p0"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p0">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</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></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:82_2_2_2_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="card_fin.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E6:82_2_2_2_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:82_2_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</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">k</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a></i>;<br/><a NAME="E8:82_2_2_2_1_1_2"/><b>then </b>
( 0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F1</font> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <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> )
 <b>by </b><i><a class="txt" href="card_fin.html#E29:82_2_2_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_2_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="afinsq_1.html#T37">AFINSQ_1:37</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E9:82_2_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XI1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">XI1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">XI</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">XI</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n1</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">n1</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> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="card_fin.html#E16:82_2_2_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E18:82_2_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="afinsq_1.html#T22">AFINSQ_1:22</a></i>;<br/><a NAME="E10:82_2_2_2_1_1_2"/><b>then </b>
<font color="Maroon">F1</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="binop_2.html#K11">*</a> <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">n1</font></span>)</span></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> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E11:82_2_2_2_1_1_2"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">F1</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> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_2_2_2_1_1_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 </span><b>by </b><i><a class="txt" href="card_fin.html#E1:82_2_2_2_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:82_2_2_2_1_1_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:82_2_2_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</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">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">XFyX</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> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<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">F2</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFyX</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_fin.html#E15:82_2_2_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E4:82_2_2_2_1_1_2_1_1"/>
<font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T7" target="_self">Th7</a>, , <a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_2_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:82_2_2_2_1_1_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:82_2_2_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <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>
 <b>by </b><i><a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFyX</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <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> )
 <b>by </b><i><a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:82_2_2_2_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</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></span>)</span>,<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> 0 )
 <b>by </b><i><a class="txt" href="card_fin.html#E12:82_2_2_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="afinsq_1.html#T40">AFINSQ_1:40</a>, </i>;<br/><b>hence </b><a NAME="E5:82_2_2_2_1_1_2_1_2"/>
<font color="Maroon">XFS</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T7" target="_self">Th7</a>, , <a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E31:82_2_2_2"/><b>then </b><i><font color="Green">E184</font></i>: 
<a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E29:82_2_2_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E32:82_2_2_2"/>
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">XI</font> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">XI</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">XI1</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Olive">n</font>
 <b>by </b><i>, <a class="txt" href="card_fin.html#E18:82_2_2_2"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E33:82_2_2_2"/><b>then </b><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XI</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XI1</font>
 <b>by </b><i>, <a class="txt" href="card_fin.html#E19:82_2_2_2"><i><font color="Green">E80</font></i></a>, </i>;<br/><a NAME="E34:82_2_2_2"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  &amp; 0 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E35:82_2_2_2"/><b>then </b>
( <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XcF</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T44">STIRL2_1:44</a></i>;<br/><a NAME="E36:82_2_2_2"/><b>then </b>
( <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XI1</font></span>)</span> &amp; <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFyX</font></span>)</span>,0 &amp; <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFyX</font></span>)</span>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFyX</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a>, <a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>;<br/><a NAME="E37:82_2_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F1</font></span>)</span> <a href="binop_2.html#K20">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XI1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFyX</font></span>)</span> &amp; <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XI1</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XI1</font> &amp; <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFyX</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XFyX</font> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F1</font></span>)</span> <a href="binop_2.html#K20">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F1</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>;<br/><a NAME="E38:82_2_2_2"/><b>then </b><i><font color="Green">E189</font></i>: 
( <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XI</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XFyX</font></span>)</span> &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">k</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><font color="Maroon">Fy</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p0"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> )
 <b>by </b><i>, <a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="txt" href="card_fin.html#E12:82_2_2_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="stirl2_1.html#T65">STIRL2_1:65</a>, </i>;<br/><a NAME="E39:82_2_2_2"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">urngFyX</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XFyX</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">urngI</font> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XI</font> &amp; <font color="Maroon">urngFyX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">urngFy</font> &amp; <font color="Maroon">urngFyX</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">urngI</font> )
 <b>by </b><i>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green">E68</font></i></a>, , <a class="txt" href="card_fin.html#E12:82_2_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_fin.html#E15:82_2_2_2"><i><font color="Green">E75</font></i></a>, , <a class="txt" href="card_fin.html#E16:82_2_2_2"><i><font color="Green">E77</font></i></a>, , </i>;<br/><a NAME="E40:82_2_2_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">urngFy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XFyX</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">Fy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">XI</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T64">CARD_2:64</a></i>;<br/><b>hence </b><a NAME="E41:82_2_2_2"/>
 <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">XFS</font>
 <b>by </b><i><a class="ref" href="card_fin.html#T5" target="_self">Th5</a>, <a class="ref" href="card_fin.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:82"/>
for <font color="Olive">k</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">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>);<br/></i>
<b>hence </b><a NAME="E4:82"/>
for <font color="Olive">Fy</font> being <a href="finset_1.html#V2">finite-yielding</a> <a href="funct_1.html#NM1">Function</a><br/> for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st  <a href="relat_1.html#K1">dom</a> <font color="Olive">Fy</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> holds <br/>for <font color="Olive">XFS</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>   st  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">XFS</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">X</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="Olive">XFS</font> holds <br/><font color="Olive">XFS</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="Olive">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> ) holds <br/> <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="Olive">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Olive">XFS</font>
 ;<br/>


</div>
