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

<b>let </b><font color="Maroon" title="c1">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>set </b><font color="Maroon" title="c2">2n</font> = 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">i</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">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">i</font>;<br/>
<b>set </b><font color="Maroon" title="c3">D</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">0,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>;<br/>
<a NAME="E1:50"/><i><font color="Green" title="E44">A1</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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">0,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:50_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">Z</font> =  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">0,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">Z</font>
;<br/>

<b>thus </b><a NAME="E3:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">Z</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">r</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">0,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span><b> such that </b><br/><a NAME="E3:50"/><i><font color="Green" title="E45">A2</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; ( 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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="catalan2.html#E1:50"><i><font color="Green" title="E44">A1</font></i></a>);<br/></i>
<a NAME="E4:50"/><i><font color="Green" title="E46">A3</font></i>: 
for <font color="Olive" title="b1">i</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">i</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">r</font> holds <br/><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</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="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</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="E1:50_2"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c5">i</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">r</font>
 ;<br/>

<a NAME="E2:50_2"/>
<font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="catalan2.html#E1:50_2"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:50_2"/>
<font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:50"/><i><font color="Green" title="E47">A5</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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">i</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">r</font> &amp; <font color="Olive" title="b2">j</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">r</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</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="E1:50_3"/><i><font color="Green" title="E48">A6</font></i>: 
( <font color="Maroon" title="c5">i</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">r</font> &amp; <font color="Maroon" title="c6">j</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">r</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> )
 ;<br/>

<b>assume </b><a NAME="E2:50_3"/>
<font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 ;<br/>

<a NAME="E3:50_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:50_3"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:50_3"/>
( <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="catalan2.html#E1:50_3"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="catalan2.html#E5:50_3"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:50_3"/><i><font color="Green" title="E50">A8</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c8">p</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> )
 <b>by </b><i><a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></i>;<br/>
<a NAME="E9:50_3"/>
( <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="catalan2.html#E1:50_3"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="catalan2.html#E5:50_3"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:50_3"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Olive" title="b1">q</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E11:50_3"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E1:50_3"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="catalan2.html#E8:50_3"><i><font color="Green" title="E50">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">Cardr</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:50"/><i><font color="Green" title="E48">A9</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">Cardr</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="c4">r</font>
 <b>and </b><br/><a NAME="E8:50"/><i><font color="Green" title="E49">A10</font></i>: 
for <font color="Olive" title="b1">i</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">i</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="c5">Cardr</font> holds <br/><font color="Maroon" title="c5">Cardr</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E9:50"/><i><font color="Green" title="E50">A11</font></i>: 
 <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="c4">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c5">Cardr</font>
 <b>from </b><i><a class="ref" href="stirl2_1.html#S7" title="STIRL2_1:sch.7">STIRL2_1:sch 7</a>(<a class="txt" href="catalan2.html#E4:50"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="catalan2.html#E5:50"><i><font color="Green" title="E47">A5</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</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>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">Cardr</font> implies  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> $1 );<br/>
<a NAME="E10:50"/><i><font color="Green" title="E51">A12</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_4"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">Cardr</font>
 ;<br/>

<a NAME="E2:50_4"/><b>then </b>
0 <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="c5">Cardr</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:50_4"/><b>then </b><i><font color="Green" title="E52">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">Cardr</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">Cardr</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="c5">Cardr</font> &amp; 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">Cardr</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E4:50_4"/><b>then </b>
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">Cardr</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T56" title="CARD_1:th.56">CARD_1:56</a></i>;<br/>
<a NAME="E5:50_4"/><b>then </b>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E6:50_4"/><b>then </b>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> 1</span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Cardr</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:50_4"/><b>then </b><i><font color="Green" title="E53">A14</font></i>: 
( <font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c5">Cardr</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> &amp; <font color="Maroon" title="c5">Cardr</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E8:50"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="catalan2.html#E3:50_4"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E8:50_4"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E9:50_4"/><b>then </b>
<font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,0
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a></i>;<br/>
<a NAME="E10:50_4"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="catalan2.html#T28" target="_self" title="CATALAN2:th.28">Th28</a></i>;<br/>
<a NAME="E11:50_4"/><b>then </b>
(  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E7:50_4"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="catalan2.html#E14"><i><font color="Green" title="E13">Lm5</font></i></a>, <a class="ref" href="newton.html#T29" title="NEWTON:th.29">NEWTON:29</a></i>;<br/>
<b>hence </b><a NAME="E12:50_4"/>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:50"/><i><font color="Green" title="E52">A15</font></i>: 
for <font color="Olive" title="b1">i</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>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</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="E1:50_5"/><i><font color="Green" title="E53">A16</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">i</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1;<br/>
<b>assume </b><a NAME="E2:50_5"/><i><font color="Green" title="E54">A17</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">Cardr</font>
 ;<br/>

<a NAME="E3:50_5"/><b>then </b>
<font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c5">Cardr</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E4:50_5"/><b>then </b><i><font color="Green" title="E55">A18</font></i>: 
( <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</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#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">Cardr</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</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="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</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="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="catalan2.html#E7:50"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="catalan2.html#E8:50"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="card_fin.html#T44" title="CARD_FIN:th.44">CARD_FIN:44</a></i>;<br/>
<a NAME="E5:50_5"/>
<font color="Maroon" title="c6">i</font> <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="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="catalan2.html#E7:50"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="catalan2.html#E2:50_5"><i><font color="Green" title="E54">A17</font></i></a></i>;<br/>
<a NAME="E6:50_5"/><b>then </b>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E7:50_5"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c6">i</font></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">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E4:50_5"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="catalan2.html#E1:50_5"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="catalan2.html#E2:50_5"><i><font color="Green" title="E54">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="catalan2.html#T32" target="_self" title="CATALAN2:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E8:50_5"/>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:50"/><i><font color="Green" title="E53">A19</font></i>: 
for <font color="Olive" title="b1">i</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>2</sub></font>[<font color="Olive" title="b1">i</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="catalan2.html#E10:50"><i><font color="Green" title="E51">A12</font></i></a>, <a class="txt" href="catalan2.html#E11:50"><i><font color="Green" title="E52">A15</font></i></a>);<br/></i>
<a NAME="E13:50"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">Cardr</font> &amp; <font color="Maroon" title="c5">Cardr</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">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> <font color="Maroon" title="c5">Cardr</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="catalan2.html#E7:50"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E14:50"/><b>then </b><i><font color="Green" title="E54">A20</font></i>: 
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c5">Cardr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E12:50"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Z</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  : (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">pN</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> ) </span>}</span> ;<br/>
<a NAME="E15:50"/><i><font color="Green" title="E55">A21</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  : (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">pN</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <font color="Maroon" title="c4">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_6"/><i><font color="Green" title="E56">A22</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  : (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">pN</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">pN</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> <b> such that </b><br/><a NAME="E3:50_6"/><i><font color="Green" title="E57">A23</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pN</font> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c8">pN</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> )
 <b>by </b><i><a class="txt" href="catalan2.html#E1:50_6"><i><font color="Green" title="E56">A22</font></i></a></i>;<br/>
<a NAME="E4:50_6"/>
<font color="Maroon" title="c8">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50_6"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="catalan2.html#T24" target="_self" title="CATALAN2:th.24">Th24</a></i>;<br/>
<a NAME="E5:50_6"/><b>then </b>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="catalan2.html#T26" target="_self" title="CATALAN2:th.26">Th26</a></i>;<br/>
<a NAME="E6:50_6"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E7:50_6"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</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="E8:50_6"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E9:50_6"/><b>then </b>
( <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span> &amp; <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span> <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="c4">r</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E10:50_6"/><b>then </b>
( <font color="Maroon" title="c8">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span> &amp; <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">pN</font></span>)</span> <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="c4">r</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50_6"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="catalan2.html#T24" target="_self" title="CATALAN2:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E11:50_6"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c4">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E3:50_6"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:50"/>
 <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="c4">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  : (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">pN</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_7"/><i><font color="Green" title="E56">A24</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c4">r</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_7"/><i><font color="Green" title="E57">A25</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">y</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="c4">r</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E1:50_7"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:50_7"/><i><font color="Green" title="E58">A26</font></i>: 
( <font color="Maroon" title="c9">i</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">r</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E3:50_7"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">i</font> 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="txt" href="catalan2.html#E5:50_7"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/>
<a NAME="E7:50_7"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c10">i</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="catalan2.html#E5:50_7"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:50_7"/><i><font color="Green" title="E59">A27</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c11">p</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E3:50_7"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></i>;<br/>
<a NAME="E10:50_7"/>
<font color="Maroon" title="c11">p</font> is    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> 
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D8" title="AFINSQ_1:def.8">AFINSQ_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E11:50_7"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  : (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">pN</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="catalan2.html#E9:50_7"><i><font color="Green" title="E59">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:50"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  : (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">pN</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E9:50"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="catalan2.html#E14:50"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="catalan2.html#E15:50"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
