<?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="E1:56"/><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">Zne</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>set </b><font color="Maroon">Ze</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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </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="E2:56"/><i><font color="Green">E53</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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </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="56_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="E1:56_1"/>
<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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 ;<br/>

<a NAME="E2:56_1"/><b>then </b>
 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; <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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:56_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">Ze</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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E4:56"/><i><font color="Green">E54</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="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="56_2"><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="E1:56_2"/>
<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/>

<a NAME="E2:56_2"/><b>then </b>
 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; <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>  )
 
;<br/>
<b>hence </b><a NAME="E3:56_2"/>
<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">Zne</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>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E6:56"/><i><font color="Green">E57</font></i>: 
 <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ze</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Zne</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><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="E1:56_3"/><i><font color="Green">E58</font></i>: 
<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/>

<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="E3:56_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E2:56"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#D8">AFINSQ_1:def 8</a></i>;<br/>
<b>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">p</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_3_1_1"/>
( <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">p</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#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <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">p</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_1_1"><b>suppose </b></a><a NAME="E1:56_3_1_1_1"/>
<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">p</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#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:56_3_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ze</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E2:56"><i><font color="Green">E53</font></i></a>, <a class="txt" href="catalan2.html#E4:56"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E3:56_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ze</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Zne</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="txt" href="catalan2.html#E4:56"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_1_2"><b>suppose </b></a><a NAME="E1:56_3_1_1_2"/>
<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">p</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/><div class="add"><a NAME="E2:56_3_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Zne</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E2:56"><i><font color="Green">E53</font></i></a>, <a class="txt" href="catalan2.html#E4:56"><i><font color="Green">E54</font></i></a></i>;<br/><b>hence </b><a NAME="E3:56_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ze</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Zne</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="txt" href="catalan2.html#E4:56"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:56_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ze</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Zne</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:56"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">Ze</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Zne</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_4"/>
<font color="Maroon">Ze</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Zne</font>
 ;<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="E3:56_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ze</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Zne</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:56_4"/><i><font color="Green">E62</font></i>: 
 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="Olive">pN</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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; <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>  )
 
<b>by </b><i><a class="txt" href="catalan2.html#E4:56"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E5:56_4"/>
 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="Olive">qN</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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; <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>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="catalan2.html#E4:56"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:56_4"/>
contradiction
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:56"/>
<font color="Maroon">Ze</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Zne</font> <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>
 
<b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="txt" href="catalan2.html#E1:56"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E9:56"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">Ze</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Zne</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">C</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="E11:56"/><i><font color="Green">E95</font></i>: 
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">Zne</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">C</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>and </b><br/><a NAME="E12:56"/><i><font color="Green">E96</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="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">C</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">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="Olive">j</font></span>)</span>,<font color="Olive">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="Olive">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="Olive">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_5_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="56_5_1_1"><b>suppose </b></a><a NAME="E1:56_5_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:56_5_1_1"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:56_5_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:56_5_1_1"/><b>then </b>
( <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K47">addnat</a>  &amp;  <a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K47">addnat</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1 )
 <b>by </b><i><a class="txt" href="catalan2.html#E6:56"><i><font color="Green">E57</font></i></a>, <a class="ref" href="stirl2_1.html#D3">STIRL2_1:def 3</a>, <a class="ref" href="binop_2.html#T5">BINOP_2:5</a></i>;<br/><b>hence </b><a NAME="E5:56_5_1_1"/>
 <a href="card_1.html#K4">card</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">C</font></span>)</span> <a href="binop_2.html#K23">+</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> 1</span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E6:56"><i><font color="Green">E57</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_5_1_2"><b>suppose </b></a><a NAME="E1:56_5_1_2"/><i><font color="Green">E98</font></i>: 
<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/><a NAME="E3:56_5_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E6:56"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:56_5_1_2"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">Ze</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> <font color="Maroon">n1</font>,<font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 )
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>hence </b><a NAME="E5:56_5_1_2"/>
 <a href="card_1.html#K4">card</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">C</font></span>)</span> <a href="binop_2.html#K23">+</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> 1</span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E2:56"><i><font color="Green">E53</font></i></a>, <a class="txt" href="catalan2.html#E4:56"><i><font color="Green">E54</font></i></a>, <a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:56"/>
(  <a href="card_1.html#K4">card</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">C</font></span>)</span> <a href="binop_2.html#K23">+</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> 1</span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> &amp;  <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">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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">C</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">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="Olive">j</font></span>)</span>,<font color="Olive">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="Olive">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="Olive">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> ) )
 <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/>


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