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

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

<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 ;<br/>

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> </span>}</span> ;<br/>
<a NAME="E3:25"/>
<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="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> </span>}</span> 
 ;<br/>

<a NAME="E2:25_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:25"/><i><font color="Green">E57</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">p</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">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E7:25"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> 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>  ;<br/>
<a NAME="E9:25"/>
 <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="E11:25"/><i><font color="Green">E58</font></i>: 
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E12:25"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

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

<a NAME="E2:25_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:25_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="afinsq_1.html#K2">dom</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="afinsq_1.html#K1">len</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="relat_1.html#T110">RELAT_1:110</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:25_2"/><b>then </b>
<a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</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#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K47">addnat</a> 
 
<b>by </b><i><a class="ref" href="stirl2_1.html#D3">STIRL2_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:25_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="binop_2.html#T5">BINOP_2:5</a></i>;<br/>


</div><b>end;</b></div>
<b>then 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/>
<b>take </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">n1</font></span>)</span>
;<br/>

<a NAME="E14:25"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:25"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/>
<a NAME="E16:25"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E17:25"/><i><font color="Green">E61</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> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_3"/><i><font color="Green">E62</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> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n1</font>
 ;<br/>

<a NAME="E2:25_3"/>
<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="E3:25_3"/><b>then </b>
( not <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:25_3"/><b>then </b><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> <font color="Maroon">n1</font></span>)</span></span>)</span> <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>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:25_3"/><b>then </b>
<font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:25_3"/><b>then </b>
( ( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> ) or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> ) &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &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="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:25_3"/><b>then </b><i><font color="Green">E95</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 &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="afinsq_1.html#T14">AFINSQ_1:14</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E8:25_3"/><b>then </b><i><font color="Green">E96</font></i>: 
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</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">n</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T43">CARD_FIN:43</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E9:25_3"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E10:25_3"/><b>then </b>
( <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="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E11:25_3"/><b>then </b>
<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="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:25_3"/><b>then </b>
( ( <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="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0 or <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="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1 ) &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></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">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</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">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> )
 
<b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E13:25_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E14:25_3"/>
 <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#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:25_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</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/>
<b>hence </b><a NAME="E16:25_3"/>
contradiction
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E1:25_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:25_4"/>
<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="E3:25_4"/><b>then </b><i><font color="Green">E99</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">n1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <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> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E4:25_4"/><b>then </b>
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</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="E5:25_4"/><b>then </b>
( <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">n</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E6:25_4"/><b>then </b>
( <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="stirl2_1.html#K7">Sum</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> 0 &amp; <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</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">n</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n1</font> &amp; <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="card_fin.html#K4">|</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font> )
 
<b>by </b><i>, , , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a>, <a class="ref" href="card_fin.html#T43">CARD_FIN:43</a></i>;<br/>
<a NAME="E7:25_4"/><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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 0 &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/>
<b>hence </b><a NAME="E8:25_4"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:25"/>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <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="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">n</font> </span>}</span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> &amp;  <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">n1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div>
