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<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="E1:28"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 ;<br/>

<b>set </b><font color="Maroon">q</font> = 1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0;<br/>
<a NAME="E2:28"/>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> &amp; 1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0 is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, </i>;<br/>
<a NAME="E3:28"/><b>then </b>
( 1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default">1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; 1 <a href="catalan2.html#K1" target="_self">--&gt;</a> 0 is <a href="catalan2.html#V1" target="_self">dominated_by_0</a> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:28"/>
<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="afinsq_1.html#K5">^</a> <font color="Maroon">p</font> is <a href="catalan2.html#V1" target="_self">dominated_by_0</a>
 <b>by </b><i>, </i>;<br/>

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</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"><span class="p3">(<span class="default"><span class="p4"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">p</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E5:28"/>
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</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"><span class="p3">(<span class="default"><span class="p4"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">p</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:28"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</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"><span class="p3">(<span class="default"><span class="p4"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">p</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">i</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="E9:28"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</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"><span class="p3">(<span class="default"><span class="p4"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">p</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E11:28"/>
 <a href="afinsq_1.html#K2">dom</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T36">AFINSQ_1:36</a></i>;<br/>
<a NAME="E12:28"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</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> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i1</font>
 
;<br/>
<a NAME="E13:28"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">p</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</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="afinsq_1.html#K5">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:28"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">p</font></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</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"><a href="afinsq_1.html#K9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i1</font></span>)</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="E15:28"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</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="Maroon">i1</font></span>)</span></span>)</span> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</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="E16:28"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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