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

<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/>

<a NAME="E1:12"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></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> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E2:12"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="catalan2.html#D1" target="_self">CATALAN2:def 1</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>assume </b><a NAME="E3:12"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span>
 ;<br/>

<a NAME="E4:12"/>
 <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0 &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:12"/>
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">k</font> <a href="catalan2.html#K1" target="_self">--&gt;</a> 0</span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>


</div>
