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

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

<b>set </b><font color="Maroon">2n</font> = 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Olive">i</font>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="catalan2.html#K3" target="_self">^omega</a> </span>)</span>;<br/>
<a NAME="E1:50"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="catalan2.html#K3" target="_self">^omega</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">Z</font> =  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="catalan2.html#K3" target="_self">^omega</a> </span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">Z</font>
;<br/>

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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">r</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="catalan2.html#K3" target="_self">^omega</a> </span>)</span><b> such that </b><br/><a NAME="E3:50"/><i><font color="Green">E53</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6">STIRL2_1:sch 6</a>();<br/></i>
<a NAME="E4:50"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is <a href="finset_1.html#V1">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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:50_2"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:50_2"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:50"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 ;<br/>

<b>assume </b><a NAME="E2:50_3"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E3:50_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:50_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:50_3"/>
( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:50_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:50_3"/>
( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">j</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:50_3"/><b>then </b>
 ex <font color="Olive">q</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">q</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:50_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="catalan2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Cardr</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:50"/><i><font color="Green">E62</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E8:50"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font> holds <br/><font color="Maroon">Cardr</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 <b>and </b><br/><a NAME="E9:50"/><i><font color="Green">E95</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Cardr</font>
 <b>from </b><i><a class="ref" href="stirl2_1.html#S7">STIRL2_1:sch 7</a>(, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font> implies  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E10:50"/><i><font color="Green">E96</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" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font>
 ;<br/>

<a NAME="E2:50_4"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Cardr</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:50_4"/><b>then </b><i><font color="Green">E97</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Cardr</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Cardr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font> &amp; 0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Cardr</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E4:50_4"/><b>then </b>
1 <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E5:50_4"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="hidden.html#R2">in</a> 1 )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E6:50_4"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">Cardr</font> <a href="stirl2_1.html#K8">.</a> 0 )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:50_4"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cardr</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <font color="Maroon">Cardr</font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 0</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E8:50_4"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E9:50_4"/><b>then </b>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,0
 
<b>by </b><i/>;<br/>
<a NAME="E10:50_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E11:50_4"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> 0 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="newton.html#T29">NEWTON:29</a></i>;<br/>
<b>hence </b><a NAME="E12:50_4"/>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:50"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="binop_2.html#K23">+</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_5"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<b>set </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>assume </b><a NAME="E2:50_5"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font>
 ;<br/>

<a NAME="E3:50_5"/><b>then </b>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E4:50_5"/><b>then </b><i><font color="Green">E132</font></i>: 
( <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="stirl2_1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">Cardr</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="card_fin.html#T44">CARD_FIN:44</a></i>;<br/>
<a NAME="E5:50_5"/>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , , <a class="txt" href="catalan2.html#E4:50"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E6:50_5"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E7:50_5"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="txt" href="catalan2.html#E3:50"><i><font color="Green">E53</font></i></a>, <a class="txt" href="catalan2.html#E4:50"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/>
<b>hence </b><a NAME="E8:50_5"/>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:50"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="catalan2.html#E1:50"><i><font color="Green">E51</font></i></a>, );<br/></i>
<a NAME="E13:50"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Cardr</font> &amp; <font color="Maroon">Cardr</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Cardr</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E14:50"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Cardr</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:50"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">pN</font> where <font color="Olive">pN</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  : (  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">pN</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">pN</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> ) </span>}</span> ;<br/>
<a NAME="E15:50"/><i><font color="Green">E135</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">pN</font> where <font color="Olive">pN</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  : (  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">pN</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">pN</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_6"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">pN</font> where <font color="Olive">pN</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  : (  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">pN</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">pN</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">pN</font> being    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a> <b> such that </b><br/><a NAME="E3:50_6"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pN</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">pN</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">pN</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> )
 <b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:50_6"/>
<font color="Maroon">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:50_6"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:50_6"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E7:50_6"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:50_6"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E9:50_6"/><b>then </b>
( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E10:50_6"/><b>then </b>
( <font color="Maroon">pN</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:50_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:50"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">pN</font> where <font color="Olive">pN</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  : (  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">pN</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">pN</font> is <a href="catalan2.html#V1" target="_self">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_7"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_7"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:50_7"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="catalan2.html#E5:50"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E7:50_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E5:50"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:50_7"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:50_7"/>
<font color="Maroon">p</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a> 
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D8">AFINSQ_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E11:50_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">pN</font> where <font color="Olive">pN</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  : (  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">pN</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">pN</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:50"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">pN</font> where <font color="Olive">pN</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  : (  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">pN</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">pN</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="txt" href="catalan2.html#E4:50"><i><font color="Green">E54</font></i></a>, <a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
