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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">Fy</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b2">X</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>for <font color="Olive" title="b3">XFS</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b3">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">X</font> &amp; ( for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b3">XFS</font> holds <br/><font color="Olive" title="b3">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Olive" title="b1">Fy</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Olive" title="b3">XFS</font>;<br/>
<a NAME="E1:82"/><i><font color="Green" title="E63">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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" title="c1">Fy</font> be  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E64">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">XFS</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:82_1"/><i><font color="Green" title="E65">A3</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E3:82_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">XFS</font> holds <br/><font color="Maroon" title="c3">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c1">Fy</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:82_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:82_1"><i><font color="Green" title="E64">A2</font></i></a>, <a class="txt" href="card_fin.html#E2:82_1"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E5:82_1"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c3">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  )
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a>, <a class="ref" href="stirl2_1.html#D3" title="STIRL2_1:def.3">STIRL2_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:82_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c3">XFS</font>
 <b>by </b><i><a class="ref" href="binop_2.html#T4" title="BINOP_2:th.4">BINOP_2:4</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:82"/><i><font color="Green" title="E64">A4</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green" title="E65">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">Fy</font> be  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E2:82_2"/><i><font color="Green" title="E66">A6</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">XFS</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:82_2"/><i><font color="Green" title="E67">A7</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E4:82_2"/><i><font color="Green" title="E68">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">XFS</font> holds <br/><font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:82_2"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font> holds <br/><font color="Olive" title="b1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:82_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font>
 ;<br/>

<a NAME="E3:82_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Fy</font> &amp; <font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:82_2_1"/>
<font color="Maroon" title="c5">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">urngFy</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">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" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="E69">A9</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:82_2_2_1"/><b>then </b>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>;<br/><a NAME="E3:82_2_2_1"/><b>then </b>
( <font color="Maroon" title="c4">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">XFS</font> &amp; <font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:82_2_2_1"/><b>then </b><i><font color="Green" title="E70">A10</font></i>: 
( <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c4">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E4:82_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:82_2_2_1"/><i><font color="Green" title="E71">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_1"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><a NAME="E7:82_2_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E6:82_2_2_1"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a></i>;<br/><a NAME="E8:82_2_2_1"/><b>then </b>
( <font color="Maroon" title="c2">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E6:82_2_2_1"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="funcop_1.html#T15" title="FUNCOP_1:th.15">FUNCOP_1:15</a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/><b>hence </b><a NAME="E9:82_2_2_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c4">XFS</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E4:82_2_2_1"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="card_fin.html#T54" target="_self" title="CARD_FIN:th.54">Th54</a></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" title="E69">A12</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:82_2_2_2"/><i><font color="Green" title="E70">A13</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Fy</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c7">Xx</font> = <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>set </b><font color="Maroon" title="c8">FyX</font> = <font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">I</font> =  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>;<br/><a NAME="E4:82_2_2_2"/>
<font color="Maroon" title="c5">urngFy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="card_fin.html#T65" target="_self" title="CARD_FIN:th.65">Th65</a></i>;<br/><a NAME="E5:82_2_2_2"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">urngFy</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">urngFyX</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E7:82_2_2_2"/>
<font color="Maroon" title="c10">urngFyX</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T58" target="_self" title="CARD_FIN:th.58">Th58</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">urngI</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E9:82_2_2_2"/><i><font color="Green" title="E71">A14</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">XFyX</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E11:82_2_2_2"/><i><font color="Green" title="E72">A15</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c12">XFyX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E12:82_2_2_2"/><i><font color="Green" title="E73">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c12">XFyX</font> holds <br/><font color="Maroon" title="c12">XFyX</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T66" target="_self" title="CARD_FIN:th.66">Th66</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">XI</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E14:82_2_2_2"/><i><font color="Green" title="E74">A17</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E15:82_2_2_2"/><i><font color="Green" title="E75">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XI</font> holds <br/><font color="Maroon" title="c13">XI</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_fin.html#T66" target="_self" title="CARD_FIN:th.66">Th66</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">XI</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> $1</span>)</span>;<br/><a NAME="E16:82_2_2_2"/><i><font color="Green" title="E76">A19</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">XI1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>, <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E18:82_2_2_2"/><i><font color="Green" title="E77">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c14">XI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="card_fin.html#E16:82_2_2_2"><i><font color="Green" title="E76">A19</font></i></a>);<br/></i><a NAME="E19:82_2_2_2"/><i><font color="Green" title="E78">A21</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">XI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">XI1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">XI1</font> = <font color="Maroon" title="c14">XI1</font> as   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7" title="AFINSQ_1:th.7">AFINSQ_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">XI1</font> = <font color="Maroon" title="c15">XI1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="txt" href="card_fin.html#E19:82_2_2_2"><i><font color="Green" title="E78">A21</font></i></a>, <a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/><b>set </b><font color="Maroon" title="c17">Fyx</font> = <font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>;<br/><a NAME="E22:82_2_2_2"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><a NAME="E23:82_2_2_2"/><b>then </b><i><font color="Green" title="E79">A22</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">XI1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">XFyX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a>, <a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a>, <a class="ref" href="subset_1.html#T55" title="SUBSET_1:th.55">SUBSET_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">XcF</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>, <font color="Maroon" title="c19">X0</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">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" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">XI1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">XFyX</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="txt" href="card_fin.html#E23:82_2_2_2"><i><font color="Green" title="E79">A22</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E26:82_2_2_2"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">XI1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">XFyX</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T29" title="AFINSQ_1:th.29">AFINSQ_1:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">F1</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">XI1</font>, <font color="Maroon" title="c21">F2</font> = <font color="Maroon" title="c12">XFyX</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/><a NAME="E28:82_2_2_2"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">XI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">XFyX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="card_fin.html#E11:82_2_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="txt" href="card_fin.html#E19:82_2_2_2"><i><font color="Green" title="E78">A21</font></i></a>, <a class="ref" href="afinsq_1.html#T37" title="AFINSQ_1:th.37">AFINSQ_1:37</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></i>;<br/><a NAME="E29:82_2_2_2"/><b>then </b><i><font color="Green" title="E80">A23</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E30:82_2_2_2"/><i><font color="Green" title="E81">A24</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">XFS</font> holds <br/><font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">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" title="c22">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:82_2_2_2_1"/><i><font color="Green" title="E82">A25</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">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" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="E83">A26</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:82_2_2_2_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c12">XFyX</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="card_fin.html#E11:82_2_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">XFyX</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">XFyX</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E12:82_2_2_2"><i><font color="Green" title="E73">A16</font></i></a>, <a class="ref" href="afinsq_1.html#T39" title="AFINSQ_1:th.39">AFINSQ_1:39</a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><a NAME="E4:82_2_2_2_1_1_1"/><b>then </b>
<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T14" title="GR_CY_1:th.14">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" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,1 &amp; <font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E4:82_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_2_1"><i><font color="Green" title="E82">A25</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_2_1_1_1"><i><font color="Green" title="E83">A26</font></i></a>, <a class="ref" href="card_fin.html#T56" target="_self" title="CARD_FIN:th.56">Th56</a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E6:82_2_2_2_1_1_1"/>
<font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:82_2_2_2_1_1_1"><i><font color="Green" title="E83">A26</font></i></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"><b>suppose </b></a><a NAME="E1:82_2_2_2_1_1_2"/><i><font color="Green" title="E83">A27</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c23">n1</font> = <font color="Maroon" title="c22">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_2"/>
(  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c22">n</font></span>)</span> &amp; <font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E4:82_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="card_fin.html#E9:82_2_2_2"><i><font color="Green" title="E71">A14</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_2_1"><i><font color="Green" title="E82">A25</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_2_1_1_2"><i><font color="Green" title="E83">A27</font></i></a>, <a class="ref" href="card_fin.html#T61" target="_self" title="CARD_FIN:th.61">Th61</a></i>;<br/><a NAME="E4:82_2_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E84">A28</font></i>: 
<font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p3">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c22">n</font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:82_2_2_2_1_1_2"/>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2"><i><font color="Green" title="E67">A7</font></i></a>, <a class="txt" href="card_fin.html#E1:82_2_2_2_1"><i><font color="Green" title="E82">A25</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/><a NAME="E6:82_2_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E85">A29</font></i>: 
( <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c23">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:82_2_2_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:82_2_2_2_1_1_2"/><b>then </b>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">F1</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c23">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E29:82_2_2_2"><i><font color="Green" title="E80">A23</font></i></a>, <a class="txt" href="card_fin.html#E6:82_2_2_2_1_1_2"><i><font color="Green" title="E85">A29</font></i></a>, <a class="ref" href="afinsq_1.html#T37" title="AFINSQ_1:th.37">AFINSQ_1:37</a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:82_2_2_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XI1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c23">n1</font> &amp; <font color="Maroon" title="c16">XI1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c23">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">XI</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c23">n1</font></span>)</span> &amp; <font color="Maroon" title="c13">XI</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c23">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c23">n1</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green" title="E74">A17</font></i></a>, <a class="txt" href="card_fin.html#E15:82_2_2_2"><i><font color="Green" title="E75">A18</font></i></a>, <a class="txt" href="card_fin.html#E18:82_2_2_2"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="afinsq_1.html#T22" title="AFINSQ_1:th.22">AFINSQ_1:22</a></i>;<br/><a NAME="E10:82_2_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c23">n1</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E11:82_2_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E86">A30</font></i>: 
<font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">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" title="c22">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> or <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 </span><b>by </b><i><a class="txt" href="card_fin.html#E6:82_2_2_2_1_1_2"><i><font color="Green" title="E85">A29</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c22">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 ;<br/><div class="add"><a NAME="E2:82_2_2_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c12">XFyX</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">XFyX</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E11:82_2_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="txt" href="card_fin.html#E12:82_2_2_2"><i><font color="Green" title="E73">A16</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">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" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E4:82_2_2_2_1_1_2"><i><font color="Green" title="E84">A28</font></i></a>, <a class="txt" href="card_fin.html#E11:82_2_2_2_1_1_2"><i><font color="Green" title="E86">A30</font></i></a>, <a class="ref" href="gr_cy_1.html#T14" title="GR_CY_1:th.14">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" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 ;<br/><div class="add"><a NAME="E2:82_2_2_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></i>;<br/><a NAME="E3:82_2_2_2_1_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">XFyX</font> &amp; <font color="Maroon" title="c22">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E11:82_2_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:82_2_2_2_1_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="card_fin.html#E9:82_2_2_2"><i><font color="Green" title="E71">A14</font></i></a>, <a class="ref" href="card_fin.html#T51" target="_self" title="CARD_FIN:th.51">Th51</a>, <a class="ref" href="afinsq_1.html#T40" title="AFINSQ_1:th.40">AFINSQ_1:40</a></i>;<br/><b>hence </b><a NAME="E5:82_2_2_2_1_1_2_1_2"/>
<font color="Maroon" title="c4">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">F2</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E4:82_2_2_2_1_1_2"><i><font color="Green" title="E84">A28</font></i></a>, <a class="txt" href="card_fin.html#E11:82_2_2_2_1_1_2"><i><font color="Green" title="E86">A30</font></i></a>, <a class="ref" href="gr_cy_1.html#T14" title="GR_CY_1:th.14">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"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c13">XI</font> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">XI</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XI1</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green" title="E74">A17</font></i></a>, <a class="txt" href="card_fin.html#E18:82_2_2_2"><i><font color="Green" title="E77">A20</font></i></a></i>;<br/><a NAME="E32:82_2_2_2"/><b>then </b><i><font color="Green" title="E82">A31</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c13">XI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c16">XI1</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green" title="E74">A17</font></i></a>, <a class="txt" href="card_fin.html#E19:82_2_2_2"><i><font color="Green" title="E78">A21</font></i></a>, <a class="ref" href="card_fin.html#T64" target="_self" title="CARD_FIN:th.64">Th64</a></i>;<br/><a NAME="E33:82_2_2_2"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><a NAME="E34:82_2_2_2"/><b>then </b>
( <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c18">XcF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp; <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c19">X0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><a NAME="E35:82_2_2_2"/><b>then </b>
( <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c20">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c16">XI1</font></span>)</span> &amp; <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c21">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c12">XFyX</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c12">XFyX</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c12">XFyX</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T14" title="GR_CY_1:th.14">GR_CY_1:14</a>, <a class="ref" href="stirl2_1.html#T47" title="STIRL2_1:th.47">STIRL2_1:47</a></i>;<br/><a NAME="E36:82_2_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c20">F1</font></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c21">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c16">XI1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c12">XFyX</font></span>)</span> &amp; <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c16">XI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c16">XI1</font> &amp; <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c12">XFyX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c12">XFyX</font> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c20">F1</font></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c21">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c20">F1</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c21">F2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T14" title="GR_CY_1:th.14">GR_CY_1:14</a></i>;<br/><a NAME="E37:82_2_2_2"/><b>then </b><i><font color="Green" title="E83">A32</font></i>: 
( <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c21">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c13">XI</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c12">XFyX</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="card_fin.html#E9:82_2_2_2"><i><font color="Green" title="E71">A14</font></i></a>, <a class="txt" href="card_fin.html#E32:82_2_2_2"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="card_fin.html#T57" target="_self" title="CARD_FIN:th.57">Th57</a>, <a class="ref" href="stirl2_1.html#T47" title="STIRL2_1:th.47">STIRL2_1:47</a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></i>;<br/><a NAME="E38:82_2_2_2"/><b>then </b>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">urngFyX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c12">XFyX</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">urngI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c13">XI</font> &amp; <font color="Maroon" title="c10">urngFyX</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">urngFy</font> &amp; <font color="Maroon" title="c10">urngFyX</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">urngI</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:82_2"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="card_fin.html#E2:82_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="card_fin.html#E3:82_2_2_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="card_fin.html#E9:82_2_2_2"><i><font color="Green" title="E71">A14</font></i></a>, <a class="txt" href="card_fin.html#E11:82_2_2_2"><i><font color="Green" title="E72">A15</font></i></a>, <a class="txt" href="card_fin.html#E12:82_2_2_2"><i><font color="Green" title="E73">A16</font></i></a>, <a class="txt" href="card_fin.html#E14:82_2_2_2"><i><font color="Green" title="E74">A17</font></i></a>, <a class="txt" href="card_fin.html#E15:82_2_2_2"><i><font color="Green" title="E75">A18</font></i></a>, <a class="ref" href="card_fin.html#T58" target="_self" title="CARD_FIN:th.58">Th58</a>, <a class="ref" href="card_fin.html#T65" target="_self" title="CARD_FIN:th.65">Th65</a></i>;<br/><a NAME="E39:82_2_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">urngFy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c12">XFyX</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c13">XI</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T64" title="CARD_2:th.64">CARD_2:64</a></i>;<br/><b>hence </b><a NAME="E40:82_2_2_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Maroon" title="c4">XFS</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E29:82_2_2_2"><i><font color="Green" title="E80">A23</font></i></a>, <a class="txt" href="card_fin.html#E30:82_2_2_2"><i><font color="Green" title="E81">A24</font></i></a>, <a class="txt" href="card_fin.html#E37:82_2_2_2"><i><font color="Green" title="E83">A32</font></i></a>, <a class="ref" href="card_fin.html#T62" target="_self" title="CARD_FIN:th.62">Th62</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" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="card_fin.html#E1:82"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="card_fin.html#E2:82"><i><font color="Green" title="E64">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:82"/>
for <font color="Olive" title="b1">Fy</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b2">X</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">X</font> holds <br/>for <font color="Olive" title="b3">XFS</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b3">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">X</font> &amp; ( for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b3">XFS</font> holds <br/><font color="Olive" title="b3">XFS</font> <a href="card_fin.html#K7" title="CARD_FIN:func.7">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Olive" title="b1">Fy</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">Fy</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K6" title="CARD_FIN:func.6">Sum</a> <font color="Olive" title="b3">XFS</font>
 ;<br/>


</div>
