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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="E1:8"/><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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1"/>
(  not 0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> or 0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1"/>
not 0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_1_1_1"/>
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_2"/>
0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_2"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &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> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:8_1_1_2"/><b>then </b>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:8_1_1_2"/><b>then </b>
( 1 <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#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; 0 <a href="hidden.html#R2">in</a> 1 )
 <b>by </b><i><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>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E5:8_1_1_2"/><b>then </b>
(  <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> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 0 <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> 1 )
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:8_1_1_2"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/><a NAME="E7:8_1_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 )
 <b>by </b><i><a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a>, <a class="ref" href="stirl2_1.html#T44">STIRL2_1:44</a></i>;<br/><a NAME="E8:8_1_1_2"/><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"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 )
 <b>by </b><i>, </i>;<br/><a NAME="E9:8_1_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K23">+</a> 0
 ;<br/><a NAME="E10:8_1_1_2"/><b>then </b>
( 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 or 2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/><b>hence </b><a NAME="E11:8_1_1_2"/>
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a> 0
 ;<br/>


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