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

<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c3">p</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp; 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">q</font> = <font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font>;<br/>
<a NAME="E2:23"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:23"/><b>then </b>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c2">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="catalan2.html#T15" target="_self" title="CATALAN2:th.15">Th15</a>, <a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E4:23"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T43" title="CARD_FIN:th.43">CARD_FIN:43</a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E5:23"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E24">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:23_1"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:23_1"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="catalan2.html#T10" target="_self" title="CATALAN2:th.10">Th10</a>, <a class="ref" href="catalan2.html#T15" target="_self" title="CATALAN2:th.15">Th15</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E4:23_1"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#D1" target="_self" title="CATALAN2:def.1">Def1</a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:23_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:23_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="catalan2.html#E3:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="card_fin.html#T43" title="CARD_FIN:th.43">CARD_FIN:43</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:23_1"/><b>then </b><i><font color="Green" title="E26">A5</font></i>: 
(  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/>
<a NAME="E7:23_1"/>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<a NAME="E8:23_1"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">s</font> = <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E10:23_1"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>
<a NAME="E11:23_1"/><b>then </b>
( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="catalan2.html#E3:23_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="catalan2.html#T12" target="_self" title="CATALAN2:th.12">Th12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E12:23_1"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E1:23"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="catalan2.html#E6:23_1"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23"/><b>then </b>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></i>;<br/>
<b>hence </b><a NAME="E7:23"/>
<font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E4:23"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funcop_1.html#T15" title="FUNCOP_1:th.15">FUNCOP_1:15</a></i>;<br/>


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