<?xml version="1.0"?>
<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">q</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><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E51</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">k</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">q</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &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">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> )
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</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">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/>
<a NAME="E4:26"/><i><font color="Green">E57</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</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">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_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="E1:26_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</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">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 ;<br/>

<a NAME="E2:26_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="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:26_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">k0</font> = <font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1;<br/>
<b>set </b><font color="Maroon">kqk</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span>;<br/>
<a NAME="E5:26"/>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</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"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:26"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K24">*</a> 0 &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K24">*</a> 1 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:26"/><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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span></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">q</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E8:26"/><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"><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> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></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">q</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0 )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:26"/><b>then </b><i><font color="Green">E59</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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</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">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</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">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E11:26"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E60</font></i>: 
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>

<a NAME="E2:26_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T30">AFINSQ_1:30</a></i>;<br/>
<a NAME="E3:26_2"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="afinsq_1.html#K5">^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><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>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:26_2"/><b>then </b><i><font color="Green">E61</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</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>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:26_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</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">k</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">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E6:26_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<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"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <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>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E7:26_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/>
<a NAME="E8:26_2"/><b>then </b><i><font color="Green">E95</font></i>: 
 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>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1_1"/>
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1_1"/><i><font color="Green">E96</font></i>: 
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:26_2_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="afinsq_1.html#T30">AFINSQ_1:30</a></i>;<br/><a NAME="E3:26_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <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"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <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"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0 )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:26_2_1_1_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"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span></span>)</span> <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"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K24">*</a> 0 )
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/><b>hence </b><a NAME="E5:26_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_1_2"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">mk</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> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1_1_2_1_1"/>
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <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> or  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1_2_1_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <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>
 ;<br/><div class="add"><a NAME="E2:26_2_1_1_2_1_1_1"/>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mk</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> )
 <b>by </b><i><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>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E3:26_2_1_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <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"><a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <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"><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"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">mk</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:26_2_1_1_2_1_1_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"><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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">mk</font></span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K24">*</a> 0 )
 <b>by </b><i>, , </i>;<br/><a NAME="E5:26_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">mk</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mk</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , , , <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><a NAME="E6:26_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T59">NAT_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_1_2_1_1_2"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">mkL</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> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> 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#T61">NAT_1:61</a></i>;<br/><a NAME="E3:26_2_1_1_2_1_1_2"/>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <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>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:26_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">mkL</font> &amp; <font color="Maroon">mkL</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:26_2_1_1_2_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> <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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">mkL</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">mkL</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mkL</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E6:26_2_1_1_2_1_1_2"/><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"><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"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</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"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">mkL</font></span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">mkL</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mkL</font> <a href="binop_2.html#K24">*</a> 1 &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K24">*</a> 0 &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="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E7:26_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:26_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:26_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:26"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</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">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>


</div>
