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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>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><a NAME="E1:23"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &amp; 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <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="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">q</font> = <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font>;<br/>
<a NAME="E2:23"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</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#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:23"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E4:23"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</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">k</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</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">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</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">k</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</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="E5:23"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:23_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</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/>
<a NAME="E3:23_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> &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">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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">k</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">k</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i>, , , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E4:23_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</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">k</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:23_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</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">k</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</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">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="card_fin.html#T43">CARD_FIN:43</a></i>;<br/>
<a NAME="E6:23_1"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</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">0</span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:23_1"/>
 <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">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E8:23_1"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <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="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/>
<a NAME="E10:23_1"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">s</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:23_1"/><b>then </b>
( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/>
<b>hence </b><a NAME="E12:23_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</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">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><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">k</font></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T36">AFINSQ_1:36</a></i>;<br/>
<b>hence </b><a NAME="E7:23"/>
<font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T15">FUNCOP_1:15</a></i>;<br/>


</div>
