<?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">p</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <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>  &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 ;<br/>

<b>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>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<a NAME="E4:24"/>
 <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">n</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="E6:24"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">n</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="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E8:24"/>
 <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> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E9:24"/><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> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E10:24"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="binop_2.html#K24">*</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:24"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<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="E13:24"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</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> <font color="Maroon">n1</font></span>)</span> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>set </b><font color="Maroon">q0</font> = 1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">q1</font> = 1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 1;<br/>
<b>set </b><font color="Maroon">qq</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span>;<br/>
<a NAME="E14:24"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</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="E15:24"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E16:24"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E17:24"/><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> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1 is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> )
 
<b>by </b><i>, , <a class="ref" href="afinsq_1.html#T14">AFINSQ_1:14</a></i>;<br/>
<a NAME="E18:24"/><b>then </b>
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</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> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0</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> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; 1 <a href="tarski.html#R1">c=</a> <font color="Maroon">n1</font> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, , <a class="ref" href="afinsq_1.html#T36">AFINSQ_1:36</a>, <a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E19:24"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1 &amp; <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#T103">RELAT_1:103</a>, <a class="ref" href="funcop_1.html#T15">FUNCOP_1:15</a></i>;<br/>
<b>hence </b><a NAME="E20:24"/>
<font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/>

<a NAME="E21:24"/>
(  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">m</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> )
 
<b>by </b><i>, , <a class="ref" href="afinsq_1.html#T27">AFINSQ_1:27</a>, <a class="ref" href="afinsq_1.html#T28">AFINSQ_1:28</a></i>;<br/>
<a NAME="E22:24"/><b>then </b>
(  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <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">1 <a href="catalan2.html#K1" target="_self">--&gt;</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>hence </b><a NAME="E23:24"/>
 <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="catalan2.html#D1" target="_self">CATALAN2:def 1</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="E24:24"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E25:24"/>
 <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> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T14">AFINSQ_1:14</a></i>;<br/>
<a NAME="E26:24"/><b>then </b>
( <font color="Maroon">n1</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">1 <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> <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#T20">AFINSQ_1:20</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E27:24"/><b>then </b>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E28:24"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n1</font> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default">1 <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">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E29:24"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <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">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</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">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> &amp; 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <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#T103">RELAT_1:103</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E30:24"/><b>then </b><i><font color="Green">E62</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">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</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">1 <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">k</font></span>)</span></span>)</span> &amp; 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K24">*</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, , </i>;<br/>
<a NAME="E31:24"/>
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">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E93</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">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:24_1"/>
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"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:24_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> <span class="p3">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> &amp; 1 <a href="binop_2.html#K23">+</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#T1">XREAL_1:1</a></i>;<br/>
<a NAME="E4:24_1"/><b>then </b>
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
;<br/>
<b>hence </b><a NAME="E5:24_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E32:24"/>
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="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div>
