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<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>assume </b><a NAME="E1:54"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">CardF</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54"/><i><font color="Green">E53</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">pN</font> where <font color="Olive">pN</font> is    <a href="catalan2.html#M2" target="_self">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="catalan2.html#K3" target="_self">^omega</a>  : ( <font color="Olive">pN</font> <a href="hidden.html#R2">in</a>  <a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font> &amp; <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"><font color="Olive">pN</font> <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>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">CardF</font>
 <b>and </b><br/><a NAME="E4:54"/><i><font color="Green">E54</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">CardF</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">CardF</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">j</font></span>)</span>,<font color="Olive">j</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p5">(<span class="default"><font color="Olive">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Olive">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">CardF</font>
;<br/>

<a NAME="E6:54"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">CardF</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E7:54"/>
(  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">CardF</font> <a href="hidden.html#R1">=</a>  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">CardF</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, </i>;<br/>

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E8:54"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E9:54"/>
( <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">j</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:54"/><b>then </b>
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>then reconsider </b><font color="Maroon">nj</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <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="E12:54"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:54"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E14:54"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E3:54"><i><font color="Green">E53</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>then </b><i><font color="Green">E60</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">nj</font></span>)</span>,<font color="Maroon">nj</font></span>)</span>

<br/>.= 
 <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">nj</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E16:54"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" target="_self">Domin_0</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalan1.html#K1">Catalan</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:54"/>
<font color="Maroon">CardF</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="catalan1.html#K1">Catalan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E1:54"><i><font color="Green">E51</font></i></a>, , <a class="txt" href="catalan2.html#E4:54"><i><font color="Green">E54</font></i></a></i>;<br/>


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