<?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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green">E51</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">W</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> ;<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="Olive">j</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> <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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Z</font> =  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>;<br/>
<a NAME="E2:52"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> 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"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></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="52_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="E1:52_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 ;<br/>

<b>set </b><font color="Maroon">N</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span> ;<br/>
<a NAME="E2:52_1"/>
<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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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> ; <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="E1:52_1_1"/><i><font color="Green">E54</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span> 
 ;<br/>

<a NAME="E2:52_1_1"/>
 ex <font color="Olive">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>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">pN</font> &amp; <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:52_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">N</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span>  as    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">N</font>
;<br/>

<b>thus </b><a NAME="E4:52_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">N</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">C</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"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green">E57</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &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">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">C</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="E5:52"/><i><font color="Green">E58</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>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:52_2"/>
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E3:52_2"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:52"/><i><font color="Green">E60</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">C</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">C</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">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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> ;<br/><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="E1:52_3"/><i><font color="Green">E61</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">C</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">C</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="E2:52_3"/>
<font color="Maroon">C</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">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E3:52_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">C</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">C</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="E5:52_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">C</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">C</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="E6:52_3"/>
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 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="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E1:52"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then 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="E8:52_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pN</font> &amp; <font color="Maroon">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 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="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  )
 ;<br/>
<a NAME="E9:52_3"/>
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">qN</font> where <font color="Olive">qN</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>  : ( <font color="Olive">qN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</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="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">qN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E1:52"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:52_3"/><b>then </b>
 ex <font color="Olive">qN</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>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">qN</font> &amp; <font color="Olive">qN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</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="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">qN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E11:52_3"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E1:52"><i><font color="Green">E51</font></i></a>, <a class="txt" href="catalan2.html#E2:52"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">CardC</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="E8:52"/><i><font color="Green">E95</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">CardC</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E9:52"/><i><font color="Green">E96</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">CardC</font> holds <br/><font color="Maroon">CardC</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">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 <b>and </b><br/><a NAME="E10:52"/><i><font color="Green">E97</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">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">CardC</font>
 <b>from </b><i><a class="ref" href="stirl2_1.html#S7">STIRL2_1:sch 7</a>(, <a class="ref" href="catalan2.html#T1" target="_self">Th1</a>);<br/></i>
<b>take </b>
<font color="Maroon">CardC</font>
;<br/>

<a NAME="E11:52"/><i><font color="Green">E98</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><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="E1:52_4"/>
<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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 ;<br/>

<b>then 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="E3:52_4"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pN</font> &amp; <font color="Maroon">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>
<b>set </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/>
<a NAME="E4:52_4"/>
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:52_4_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 ;<br/>

<a NAME="E2:52_4_1"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E3:52_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:52_4"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">M</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:52_4"/><i><font color="Green">E131</font></i>: 
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>
<a NAME="E9:52_4"/>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:52_4"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>then reconsider </b><font color="Maroon">Sum1</font> = <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<b>consider </b><font color="Maroon">q</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="E13:52_4"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">pN</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<a NAME="E14:52_4"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">pN</font> <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">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E4:52"><i><font color="Green">E57</font></i></a>, </i>;<br/>
<a NAME="E15:52_4"/><b>then </b>
<font color="Maroon">m</font> <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">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E16:52_4"/><b>then </b>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">Sum1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">Sum1</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E17:52_4"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">Sum1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E18:52_4"/><b>then </b>
<font color="Maroon">Sum1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E19:52_4"/><b>then </b>
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Sum1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Sum1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">qN</font> where <font color="Olive">qN</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>  : ( <font color="Olive">qN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">Sum1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">qN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E20:52_4"/><b>then </b>
( <font color="Maroon">pN</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Sum1</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Sum1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="catalan2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E21:52_4"/>
<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">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:52"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">C</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><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="E1:52_5"/>
<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">C</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_5"/><i><font color="Green">E133</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">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:52_5"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> 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="catalan2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:52_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</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="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T3" target="_self">Th3</a></i>;<br/>
<b>then 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="E9:52_5"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pN</font> &amp; <font color="Maroon">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</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="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E10:52_5"/>
2 <a href="binop_2.html#K24">*</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#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E11:52_5"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">pN</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  )
 
<b>by </b><i><a class="txt" href="catalan2.html#E4:52"><i><font color="Green">E57</font></i></a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:52_5"/>
<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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="catalan2.html#E4:52"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:52"/>
(  <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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">CardC</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">CardC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="txt" href="catalan2.html#E1:52"><i><font color="Green">E51</font></i></a>, <a class="txt" href="catalan2.html#E2:52"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>

<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="E14:52"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<b>set </b><font color="Maroon">P</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</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="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span> ;<br/>
<a NAME="E15:52"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:52"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</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">m</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E17:52"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E18:52"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</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">n</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</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">m</font> <a href="binarith.html#K3">-'</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> <font color="Maroon">m</font> <a href="binop_2.html#K10">-</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="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E19:52"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E20:52"/><b>then </b>
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</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>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> &amp; 2 <a href="binop_2.html#K24">*</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="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">pN</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  ) </span>}</span>  &amp; <font color="Maroon">CardC</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E1:52"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E21:52"/>
<font color="Maroon">CardC</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p5">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E4:52"><i><font color="Green">E57</font></i></a>, </i>;<br/>


</div>
