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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">q</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">n</font> where <font color="Olive" title="b2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">q</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c2">q</font> is <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font> )
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c4">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E28">A4</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:26_1"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E1:26_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:26_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c5">k0</font> = <font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">k1</font> = <font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1;<br/>
<b>set </b><font color="Maroon" title="c7">kqk</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span>;<br/>
<a NAME="E5:26"/>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/>
<a NAME="E6:26"/><b>then </b>
(  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E10"><i><font color="Green" title="E10">Lm2</font></i></a>, <a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E2:26"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E8:26"/><b>then </b>
( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>
<a NAME="E9:26"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:26"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/>
<a NAME="E11:26"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">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" title="E30">A7</font></i>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:26_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>;<br/>
<a NAME="E3:26_2"/><b>then </b>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:26_2"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> )
 
<b>by </b><i><a class="txt" href="#E9:26"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/>
<a NAME="E5:26_2"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E6:26_2"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E7:26_2"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">M</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/>
<a NAME="E8:26_2"/><b>then </b><i><font color="Green" title="E34">A11</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<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="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">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" title="E35">A12</font></i>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:26_2_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</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" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E1:26_2_1_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="catalan2.html#E9"><i><font color="Green" title="E9">Lm1</font></i></a>, <a class="ref" href="catalan2.html#T2" target="_self" title="CATALAN2:th.2">Th2</a></i>;<br/><a NAME="E4:26_2_1_1_1"/><b>then </b>
(  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E5:26_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green" title="E10">Lm2</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/><b>hence </b><a NAME="E5:26_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:26_2"><i><font color="Green" title="E34">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">mk</font> = <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><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="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> or  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">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" title="E35">A13</font></i>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:26_2_1_1_2_1_1_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">mk</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">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" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c9">mk</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:26_2_1_1_2_1_1_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="catalan2.html#T2" target="_self" title="CATALAN2:th.2">Th2</a>, <a class="ref" href="catalan2.html#T3" target="_self" title="CATALAN2:th.3">Th3</a></i>;<br/><a NAME="E4:26_2_1_1_2_1_1_1"/><b>then </b>
(  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c9">mk</font></span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E6:26_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green" title="E10">Lm2</font></i></a>, <a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/><a NAME="E5:26_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">mk</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">mk</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="#E2:26"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="#E8:26_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="catalan2.html#T6" target="_self" title="CATALAN2:th.6">Th6</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</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#T19" title="NAT_1:th.19">NAT_1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">mkL</font> = <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><a NAME="E3:26_2_1_1_2_1_1_2"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="#E1:26_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="#E4:26_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:26_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">mkL</font> &amp; <font color="Maroon" title="c10">mkL</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">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" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c10">mkL</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c10">mkL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">mkL</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green" title="E9">Lm1</font></i></a>, <a class="ref" href="catalan2.html#T3" target="_self" title="CATALAN2:th.3">Th3</a></i>;<br/><a NAME="E6:26_2_1_1_2_1_1_2"/><b>then </b>
(  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c10">mkL</font></span>)</span></span>)</span> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> 1</span>)</span> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Maroon" title="c10">mkL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">mkL</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp;  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="catalan2.html#K1" title="CATALAN2:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c8">M</font> )
 <b>by </b><i><a class="txt" href="#E6:26_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="#E8:26_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="catalan2.html#E10"><i><font color="Green" title="E10">Lm2</font></i></a>, <a class="txt" href="catalan2.html#E13"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E7:26_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:26"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E1:26_2"><i><font color="Green" title="E30">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:26"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="card_fin.html#K4" title="CARD_FIN:func.4">|</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
