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

<b>let </b><font color="Maroon" title="c1">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><b>that </b><br/><a NAME="E1:25"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> <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">0,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green" title="E26">A2</font></i>: 
not <font color="Maroon" title="c1">p</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <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>  : 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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> </span>}</span> ;<br/>
<a NAME="E3:25"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <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>  : 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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <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>  : 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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> </span>}</span> 
 ;<br/>

<a NAME="E2:25_1"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:25_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <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>  : 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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c4">k</font> being    <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> such that </b><br/><a NAME="E6:25"/><i><font color="Green" title="E27">A4</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> &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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">k</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E1:25"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="catalan2.html#E2:25"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="catalan2.html#D1" target="_self" title="CATALAN2:def.1">Def1</a></i>;<br/>
<a NAME="E7:25"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E6:25"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">M</font> = <font color="Maroon" title="c3">M</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E9:25"/>
 <a href="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<b>by </b><i><a class="ref" href="henmodel.html#D1" title="HENMODEL:def.1">HENMODEL:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">n</font> being    <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> such that </b><br/><a NAME="E11:25"/><i><font color="Green" title="E28">A5</font></i>: 
(  <a href="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> &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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">n</font> )
 ;<br/>
<a NAME="E12:25"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:25_2"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<a NAME="E3:25_2"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="relat_1.html#T110" title="RELAT_1:th.110">RELAT_1:110</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:25_2"/><b>then </b>
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a> 
 
<b>by </b><i><a class="ref" href="stirl2_1.html#D3" title="STIRL2_1:def.3">STIRL2_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:25_2"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E11:25"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="binop_2.html#T5" title="BINOP_2:th.5">BINOP_2:5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">n1</font> = <font color="Maroon" title="c6">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</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#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<b>take </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span>
;<br/>

<a NAME="E14:25"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E6:25"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<a NAME="E15:25"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E11:25"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="henmodel.html#D1" title="HENMODEL:def.1">HENMODEL:def 1</a></i>;<br/>
<a NAME="E16:25"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E6:25"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:25"/><i><font color="Green" title="E31">A8</font></i>: 
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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_3"/><i><font color="Green" title="E32">A9</font></i>: 
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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">n1</font>
 ;<br/>

<a NAME="E2:25_3"/>
<font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</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="E3:25_3"/><b>then </b>
( not <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E11:25"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="catalan2.html#E16:25"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="henmodel.html#D1" title="HENMODEL:def.1">HENMODEL:def 1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:25_3"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
( 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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n1</font> &amp; <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:25_3"/><b>then </b>
<font color="Maroon" title="c7">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font>
 
<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="E6:25_3"/><b>then </b>
( ( ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> ) or <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> ) &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:25_3"/><b>then </b><i><font color="Green" title="E34">A11</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c7">n1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a>, <a class="ref" href="card_1.html#T56" title="CARD_1:th.56">CARD_1:56</a></i>;<br/>
<a NAME="E8:25_3"/><b>then </b><i><font color="Green" title="E35">A12</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</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="E9:25_3"/>
( <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E7:25_3"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E10:25_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<a NAME="E11:25_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">0,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:25"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:25_3"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</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">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> &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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:25_3"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="catalan2.html#E4:25_3"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a>, <a class="txt" href="catalan2.html#E14"><i><font color="Green" title="E13">Lm5</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E13:25_3"/><b>then </b><i><font color="Green" title="E36">A13</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E14:25_3"/>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</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"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E8:25_3"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/>
<a NAME="E15:25_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">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#E11:25"><i><font color="Green" title="E28">A5</font></i></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="E16:25_3"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E13:25_3"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:25"/>
<font color="Maroon" title="c1">p</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_4"/><i><font color="Green" title="E32">A14</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:25_4"/>
<font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</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="E3:25_4"/><b>then </b><i><font color="Green" title="E33">A15</font></i>: 
( <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c7">n1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E16:25"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="card_1.html#T56" title="CARD_1:th.56">CARD_1:56</a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/>
<a NAME="E4:25_4"/><b>then </b>
( <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a>, <a class="ref" href="card_1.html#T56" title="CARD_1:th.56">CARD_1:56</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:25_4"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</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> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n1</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="txt" href="catalan2.html#E3:25_4"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E6:25_4"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &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">0</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> 0 &amp; <font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c7">n1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E14"><i><font color="Green" title="E13">Lm5</font></i></a>, <a class="txt" href="catalan2.html#E1:25"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="catalan2.html#E1:25_4"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="catalan2.html#E3:25_4"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a>, <a class="ref" href="card_fin.html#T43" title="CARD_FIN:th.43">CARD_FIN:43</a></i>;<br/>
<a NAME="E7:25_4"/><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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c6">n</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"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0 &amp; <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n1</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>, <a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:25_4"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E11:25"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="catalan2.html#E17:25"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:25"/>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span></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>  <a href="henmodel.html#K1" title="HENMODEL:func.1">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <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>  : 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="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> </span>}</span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c1">p</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c7">n1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="catalan2.html#E11:25"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="catalan2.html#E15:25"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="catalan2.html#E17:25"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="catalan2.html#E6:25"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div>
