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

<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>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>assume </b><a NAME="E1:39"/><i><font color="Green">E51</font></i>: 
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">p</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  holds <br/> ex <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Olive">r2</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Olive">r1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Olive">r2</font> &amp; ( for <font color="Olive">o</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">o</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Olive">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">o</font></span>)</span> ) );<br/>
<b>set </b><font color="Maroon">CH</font> =  <a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0;<br/>
<b>set </b><font color="Maroon">Z</font> =  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<a NAME="E2:39"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:39"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E4:39"/><b>then </b>
( 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> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="xreal_1.html#T33">XREAL_1:33</a></i>;<br/>
<a NAME="E5:39"/><b>then </b>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:39"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> =  <a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>,1,0 as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_fin.html#T18">CARD_FIN:18</a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>
<a NAME="E8:39"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then 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="E4:39_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &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="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &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> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="card_fin.html#T46">CARD_FIN:46</a></i>;<br/>
<a NAME="E5:39_1"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:39_1"/><b>then </b>
not <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:39_1"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">q</font> = <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<a NAME="E9:39_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E10:39_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</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="E12:39_1"/><i><font color="Green">E59</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> <font color="Olive">i</font> ) &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E13:39_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">r1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, </i>;<br/>
<a NAME="E14:39_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r2</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="E16:39_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font>
 <b>by </b><i/>;<br/>
<a NAME="E17:39_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:39_1"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E19:39_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E20:39_1"/><b>then </b><i><font color="Green">E95</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E21:39_1"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E22:39_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T43">CARD_FIN:43</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E23:39_1"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E24:39_1"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_2.html#K23">+</a> 1 &amp; <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i>, , , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E25:39_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E26:39_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">R</font> = <font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font>;<br/>

<a NAME="E27:39_1"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r2</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , , <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E28:39_1"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E29:39_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T28">AFINSQ_1:28</a></i>;<br/>
<a NAME="E30:39_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:39_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E32:39_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T29">AFINSQ_1:29</a></i>;<br/>
<b>hence </b><a NAME="E33:39_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="card_fin.html#T46">CARD_FIN:46</a></i>;<br/>

<b>thus </b><a NAME="E34:39_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">R</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p'</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  )
 ;<br/>

<b>take </b>
<font color="Maroon">r1</font>
;<br/>

<b>take </b>
<font color="Maroon">r2</font>
;<br/>

<b>set </b><font color="Maroon">q'</font> = <font color="Maroon">p'</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>thus </b><a NAME="E2:39_1_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, , , , <a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E3:39_1_1"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:39_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E2:39_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E3:39_1_1_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">o</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_1_1"/>
<font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="catalan2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">W</font><b> such that </b><br/><a NAME="E10:39"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E11:39"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_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="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39_2"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 <b>and </b><br/><a NAME="E2:39_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then 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="E5:39_2"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &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="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &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> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="card_fin.html#T46">CARD_FIN:46</a></i>;<br/>
<a NAME="E6:39_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, </i>;<br/>
<a NAME="E7:39_2"/><b>then </b>
not <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:39_2"/><i><font color="Green">E136</font></i>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">K</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="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">pk</font> = <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r2</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="E12:39_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E13:39_2"/><i><font color="Green">E140</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font> )
 <b>and </b><br/><a NAME="E14:39_2"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E15:39_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:39_2"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="card_fin.html#T46">CARD_FIN:46</a></i>;<br/>
<a NAME="E18:39_2"/>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, </i>;<br/>
<a NAME="E19:39_2"/><b>then </b>
not <font color="Maroon">q</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E21:39_2"/><i><font color="Green">E144</font></i>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">M</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">q</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">qk</font> = <font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">s1</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">s2</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="E24:39_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">s2</font>
 <b>and </b><br/><a NAME="E25:39_2"/><i><font color="Green">E149</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">s2</font> )
 <b>and </b><br/><a NAME="E26:39_2"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">s1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<b>assume </b><a NAME="E27:39_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E28:39_2"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s2</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r2</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/>
<a NAME="E29:39_2"/><b>then </b><i><font color="Green">E152</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">s2</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , , <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E31:39_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="txt" href="catalan2.html#E8:39"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="afinsq_1.html#T11">AFINSQ_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2_1_1"/>
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> or <font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> or <font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1"><b>suppose </b></a><a NAME="E1:39_2_1_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_2_1_1_1_1_1_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:39_2_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, , <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E3:39_2_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">s1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, , , , , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:39_2_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">s1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">s2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:39_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="catalan2.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_2_1_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:39_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">s2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <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>, , <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="afinsq_1.html#T22">AFINSQ_1:22</a></i>;<br/><b>hence </b><a NAME="E3:39_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="ref" href="catalan2.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_2"><b>suppose </b></a><a NAME="E1:39_2_1_1_2"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><a NAME="E2:39_2_1_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:39_2_1_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:39_2_1_1_2"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E5:39_2_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">s2</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> )
 <b>by </b><i>, , <a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:39_2_1_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:39_2_1_1_2"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s2</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">s2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/><a NAME="E8:39_2_1_1_2"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E9:39_2_1_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/><a NAME="E10:39_2_1_1_2"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E11:39_2_1_1_2"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E12:39_2_1_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_2_1_1_2_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:39_2_1_1_2_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, , <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:39_2_1_1_2_1"/><b>then </b><i><font color="Green">E161</font></i>: 
( <font color="Maroon">s1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font> )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E4:39_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:39_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, , </i>;<br/>
<a NAME="E6:39_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E7:39_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E13:39_2_1_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#T11">AFINSQ_1:11</a></i>;<br/><a NAME="E14:39_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/><a NAME="E15:39_2_1_1_2"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> )
 ;<br/><a NAME="E16:39_2_1_1_2"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, </i>;<br/><b>hence </b><a NAME="E17:39_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_3"><b>suppose </b></a><a NAME="E1:39_2_1_1_3"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><a NAME="E2:39_2_1_1_3"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:39_2_1_1_3"/><b>then </b><i><font color="Green">E163</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font>
 <b>by </b><i>, <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:39_2_1_1_3"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E5:39_2_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, , <a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:39_2_1_1_3"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:39_2_1_1_3"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r2</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/><a NAME="E8:39_2_1_1_3"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E9:39_2_1_1_3"/><b>then </b><i><font color="Green">E165</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/><a NAME="E10:39_2_1_1_3"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E11:39_2_1_1_3"/><b>then </b><i><font color="Green">E166</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E12:39_2_1_1_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_2_1_1_3_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:39_2_1_1_3_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s1</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:39_2_1_1_3_1"/><b>then </b><i><font color="Green">E168</font></i>: 
( <font color="Maroon">s1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">s1</font> )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E4:39_2_1_1_3_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">s1</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:39_2_1_1_3_1"/><b>then </b>
( <font color="Maroon">s1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, , </i>;<br/>
<a NAME="E6:39_2_1_1_3_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E7:39_2_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E13:39_2_1_1_3"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#T11">AFINSQ_1:11</a></i>;<br/><a NAME="E14:39_2_1_1_3"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/><a NAME="E15:39_2_1_1_3"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> )
 ;<br/><a NAME="E16:39_2_1_1_3"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E17:39_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E33:39_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:39"/><i><font color="Green">E170</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:39_3"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_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="E1:39_3_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</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="E3:39_3_1"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &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="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &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="ref" href="card_fin.html#T46">CARD_FIN:46</a></i>;<br/>
<b>set </b><font color="Maroon">M</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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> </span>}</span> ;<br/>
<a NAME="E4:39_3_1"/><i><font color="Green">E173</font></i>: 
<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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_3_1_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> </span>}</span> 
 ;<br/>

<a NAME="E2:39_3_1_1"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:39_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:39_3_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:39_3_1"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:39_3_1"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E8:39_3_1"/><b>then </b>
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="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<a NAME="E9:39_3_1"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E11:39_3_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">mm</font> =  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font>;<br/>
<a NAME="E1:39_3_1_2"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:39_3_1_2"/><i><font color="Green">E177</font></i>: 
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">o</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">m1</font> = <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span> <a href="xcmplx_0.html#K6">-</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="E5:39_3_1_2"/>
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E6:39_3_1_2"/><b>then </b>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E7:39_3_1_2"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:39_3_1_2"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> &amp; <font color="Maroon">m1</font> <a href="tarski.html#R1">c=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E9:39_3_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m1</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">m1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">m1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">m1</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T43">CARD_FIN:43</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E10:39_3_1_2"/><b>then </b><i><font color="Green">E179</font></i>: 
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">m1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">m1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">m1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:39_3_1_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:39_3_1_2"/><b>then </b>
not <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<a NAME="E13:39_3_1_2"/><b>then </b>
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="Maroon">m1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 
;<br/>
<a NAME="E14:39_3_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">m1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E15:39_3_1_2"/><b>then </b>
( 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> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m1</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:39_3_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> &amp;  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T27">NAT_1:27</a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>hence </b><a NAME="E17:39_3_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:39_3_1"/><i><font color="Green">E180</font></i>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 ;<br/>
<b>set </b><font color="Maroon">k1</font> = <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E15:39_3_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<a NAME="E16:39_3_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E17:39_3_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</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="E19:39_3_1"/><i><font color="Green">E182</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 <b>and </b><br/><a NAME="E20:39_3_1"/><i><font color="Green">E183</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> <font color="Olive">i</font>
 <b>and </b><br/><a NAME="E21:39_3_1"/><i><font color="Green">E184</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Maroon">r</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E22:39_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, </i>;<br/>
<a NAME="E23:39_3_1"/><b>then </b><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">rq</font> = <font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>;<br/>
<a NAME="E24:39_3_1"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E25:39_3_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E26:39_3_1"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E27:39_3_1"/><b>then </b><i><font color="Green">E188</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E28:39_3_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T28">AFINSQ_1:28</a></i>;<br/>
<a NAME="E29:39_3_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E30:39_3_1"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E31:39_3_1"/><b>then </b><i><font color="Green">E189</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="afinsq_1.html#T29">AFINSQ_1:29</a></i>;<br/>
<a NAME="E32:39_3_1"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</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">r</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i>, , , , , <a class="txt" href="catalan2.html#E1:39"><i><font color="Green">E51</font></i></a>, <a class="txt" href="catalan2.html#E8:39"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E33:39_3_1"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E34:39_3_1"/><b>then </b><i><font color="Green">E191</font></i>: 
( <font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 &amp; not <font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , , <a class="ref" href="card_fin.html#T46">CARD_FIN:46</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E35:39_3_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0 &amp; not <font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E36:39_3_1"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E38:39_3_1"/><i><font color="Green">E193</font></i>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">K</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"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">rqj</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">j1</font> = <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r2</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="E41:39_3_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E42:39_3_1"/><i><font color="Green">E197</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font> )
 <b>and </b><br/><a NAME="E43:39_3_1"/><i><font color="Green">E198</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E44:39_3_1"/><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:39_3_1_3"/>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">r</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E2:39_3_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
;<br/>
<a NAME="E3:39_3_1_3"/><b>then </b><i><font color="Green">E200</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<a NAME="E4:39_3_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E5:39_3_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, , <a class="txt" href="catalan2.html#E8:39"><i><font color="Green">E53</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E6:39_3_1_3"/><b>then </b><i><font color="Green">E201</font></i>: 
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">r</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:39_3_1_3"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a>, </i>;<br/>
<a NAME="E8:39_3_1_3"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">i</font> )
 
;<br/>
<a NAME="E9:39_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E10:39_3_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:39_3_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
;<br/>
<a NAME="E12:39_3_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:39_3_1_3"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:39_3_1"/>
<font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:39_3_1_4"/>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r2</font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/>
<a NAME="E2:39_3_1_4"/><b>then </b><i><font color="Green">E202</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:39_3_1_4_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_3_1_4_1_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_3_1_4_1_1_1_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font>
 ;<br/><div class="add"><a NAME="E2:39_3_1_4_1_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:39_3_1_4_1_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:39_3_1_4_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E5:39_3_1_4_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">r1</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:39_3_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_3_1_4_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font>
 ;<br/><div class="add"><a NAME="E2:39_3_1_4_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="afinsq_1.html#T22">AFINSQ_1:22</a>, , <a class="txt" href="catalan2.html#E8:39"><i><font color="Green">E53</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:39_3_1_4_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_3_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_3_1_4"/>
<font color="Maroon">r1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="afinsq_1.html#T11">AFINSQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:39_3_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E47:39_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E13:39"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:39"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">W</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, , <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:39"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,1,0</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1">Choose</a> <font color="Maroon">n</font>,<font color="Maroon">m</font>,1,0</span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
