<?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>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>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">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">j</font></span>)</span>,<font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E1:57"/><i><font color="Green">E51</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> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><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="E1:57_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E2:57_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span>]
 
;<br/>
<b>hence </b><a NAME="E3:57_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>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="E3:57"/><i><font color="Green">E53</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E4:57"/><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> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6">STIRL2_1:sch 6</a>();<br/></i>
<a NAME="E5:57"/><i><font color="Green">E57</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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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="E1:57_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:57_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="catalan2.html#E3:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:57_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:57_2"/>
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E1:57"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_3_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_1"><b>suppose </b></a><a NAME="E1:57_3_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:57_3_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_3_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="catalan2.html#E4:57"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E3:57_3_1_1_1"/>
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="txt" href="catalan2.html#E4:57"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:57_3_1_1_1"/>
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="catalan2.html#E3:57"><i><font color="Green">E53</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:57_3_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1
 <b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/><a NAME="E4:57_3_1_1"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 1 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="txt" href="catalan2.html#E3:57"><i><font color="Green">E53</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E5:57_3_1_1"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_2"><b>suppose </b></a><a NAME="E1:57_3_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><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>for <font color="Olive">q</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; ( 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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span> ) holds <br/> <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E3:57_3_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_3_1_2_1"/><i><font color="Green">E62</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> 1 &amp; ( 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="xxreal_0.html#R1">&lt;=</a> 0 holds <br/><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span> ) )
 ;<br/>

<b>set </b><font color="Maroon">CHn</font> = <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">CHn1</font> = <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n1</font>;<br/>
<b>set </b><font color="Maroon">CHn'1</font> = <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">2CHn'1</font> = <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="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">2CHn</font> = <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="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>;<br/>
<a NAME="E2:57_3_1_2_1"/>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2 <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; 2 <a href="binop_2.html#K24">*</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> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2 &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:57_3_1_2_1"/><b>then </b>
( <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="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</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> <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> &amp; <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="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n1</font></span>)</span> &amp; <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font> <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</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="binop_2.html#K23">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</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="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 0</span>)</span>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</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="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K10">-</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n1</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E4:57"><i><font color="Green">E54</font></i></a>, <a class="ref" href="newton.html#T30">NEWTON:30</a>, <a class="ref" href="newton.html#T32">NEWTON:32</a>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:57_3_1_2_1"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</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="binop_2.html#K23">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> 0 &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> 0 &amp; <font color="Maroon">q</font> <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">q</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<b>hence </b><a NAME="E5:57_3_1_2_1"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:57_3_1_2"/><i><font color="Green">E93</font></i>: 
for <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="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><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="E1:57_3_1_2_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">j</font>]
 ;<br/>

<b>set </b><font color="Maroon">j1</font> = <font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>let </b><font color="Maroon">q</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:57_3_1_2_2"/><i><font color="Green">E96</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; ( 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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span> ) )
 ;<br/>

<b>set </b><font color="Maroon">CH1</font> = <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="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">CH2</font> = <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="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<a NAME="E3:57_3_1_2_2"/><i><font color="Green">E97</font></i>: 
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">j</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:57_3_1_2_2_1"/>
<font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E2:57_3_1_2_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E3:57_3_1_2_2_1"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:57_3_1_2_2_1"/><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="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1_2_2_1_1"><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="E1:57_3_1_2_2_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:57_3_1_2_2_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</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="E3:57_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</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="E4:57_3_1_2_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:57_3_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:57_3_1_2_2_1"/>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</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="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:57_3_1_2_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:57_3_1_2_2_1"/>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">j</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">CH3</font> = <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">CH4</font> = <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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n1</font>;<br/>
<a NAME="E4:57_3_1_2_2"/><i><font color="Green">E131</font></i>: 
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:57_3_1_2_2"/>
( <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font></span>)</span> &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:57_3_1_2_2"/><b>then </b>
<font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<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">q</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">j</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">q</font> <a href="stirl2_1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="newton.html#K9">choose</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="catalan2.html#E5:57"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T32">NEWTON:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T32">NEWTON:32</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E8:57_3_1_2_2"/>
<font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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="E9:57_3_1_2_2"/><b>then </b>
<font color="Maroon">j</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">q</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="catalan2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:57_3_1_2_2"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</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"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">j</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">q</font> <a href="stirl2_1.html#K8">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T44">CARD_FIN:44</a>, <a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<b>hence </b><a NAME="E11:57_3_1_2_2"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:57_3_1_2"/>
for <font color="Olive">j</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">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="catalan2.html#E3:57"><i><font color="Green">E53</font></i></a>, <a class="txt" href="catalan2.html#E4:57"><i><font color="Green">E54</font></i></a>);<br/></i><b>hence </b><a NAME="E6:57_3_1_2"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:57"/>
(  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; ( 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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">i</font></span>)</span>,<font color="Maroon">n</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, </i>;<br/>


</div>
